I think that *technically* this could be allowed, and I sort of wish it could be, as well as some other mutually agreeable flexibility. We ask our AP to be home a reasonable amount of time (8 hours) before her next shift to be well rested and able to safely drive our kids. So, at least in our case, I feel that I need to give the AP an equal amount of time between shifts, even if she's most likely going to be sleeping.

Not sure which agency the OP is with ... Here is the language in the CCAP agreement about hours.

"Host Family agrees that the au pair hours and work schedule will be limited to forty five (45) hours per week with a maximum of ten (10) hours per day and no more than five and one half (5.5) days per week. Host Family agrees that the au pair will have one (1) full weekend (Friday evening to Monday morning) off duty per month and one and one half (1.5) continuous days off per week. The au pair will also receive two (2) calendar weeks (14 days) of paid vacation, to be taken at mutually agreed upon times. Host Family also understands that asking the au pair to work additional hours and/or offering additional compensation for additional hours is a violation of the U.S. Department of State regulations. Host Family understands that violating any part of the working schedule rules will be cause for program termination."

The agreement is silent on whether a gap of X hours between shifts is required. So, based on the language of the agreement, I am interpreting that the OP can TECHNICALLY do this without violating the agreement. It is a loop hole in the agreement. Actually, paying extra babysitting dollars is a clear violation, while a continuous 20 hour shift spanning 2 days (with no more than 10 hours each day) is not a clear violation of the agreement but doing this is not in the spirit of the program and will breed resentment.

Better to have the AP work shifts that have a reasonable gap of hours in-between and then, have grandparents stay overnight or have HK sleep over at friends and avoid a potential resentment situation altogether.

yes, pick awkwardness over resentment. I learned this the hard way. I offered for neighbor to stay overnight, AP opted not to do this because "oh, I will just be asleep anyway so no problem" and then I got to deal with her resentment later.

your 19 hour stretch is a 19 hour stretch and those HPs implying that you can count it as two 10 hour days, don't get the rules of the program. If AP goes to LCC about it, LCC will not support your rationale.

I'd pick possible awkwardness over resentment. Keep her regular schedule (or extend by a bit) and have grandparents come sleep over (8-8). Problem solved.

Goodness. I should be away from the keyboard at this time of the day.
"Or we might schedule her from 3pm to 1am and let her have the morning off if we needed her in the afternoon afternoon (usually for us covering mornings is easier than having somebody be home in the afternoon)." should read "Or we might schedule her from 3pm to 1am and let her have the following morning off if we needed her in the afternoon."

if I schedule from 3pm-12am on Tuesday and 12am-10am on Wednesday

I agree with PP, that's not two 10 hr shifts, that's one 19 hr shift.

Personally, I am not a big fan of overnight shifts because they are so difficult to schedule if you want to stick to the rules. The best/only option really is to count back 10 hrs from when they can be dropped of at school, ie. kids need to / can be at school by 8 am, AP starts working at 10 pm the previous night and provides overnight care. But that of course requires somebody else to pick them up at school and watch them until AP starts working.
Sometimes it might be easier just to have them spend the night with a friend they go to school with or hire an overnight sitter and have the AP do the rest. Dumping them on granny and grandpa or on aunty Larla is a great option as well, if you have family in the area. And if you do, you could ask one of them to spend the night.

You need to provide your AP with a real break (ie. a break in which she can leave the house, without the children) somewhere in your scheduling. Just going on your 3pm to 10am timeframe above, I assume what you could do is have your AP work her usual morning shift on Tuesday (say 6am to 10am), then an afternoon shift (3pm to 6/7/8pm), hire a babysitter from 6/7/8pm to 12am and then have AP do nightshift from 12am to 10am on Wednesday, she would then of course need to be off until Thursday because she has already worked her 10 hrs max on Wednesday (so you couldn't count the time off on Wednesday agains her 1 1/2 days off that week).

Our definition of "day" is both calendar day (ie. not more than 10 hrs between 12.00am and 11.59pm on Monday) as well as not more than 10 consecutive hrs in a shift, even if a shift spans over two calendar days.
If we asked AP for a night of babysitting she'd not be working for more than 10 hrs from the time she started working that day until we came home and not more than 10 hrs the day her shift ran into (think 7am to 10am and 6pm to 1am on day 1 [10 hrs] and then 7am to 10am and 3pm to no longer than 9pm on day 2 [1 hr from 12 am to 1 am, 3 hrs from 7am to 10am and 6 hrs from 3pm to 9pm]). By doing that we do count the hour from 12am to 1am twice (for both day 1 and day 2) but as we pay a fixed rate and not hourly what does it really matter. Or we might schedule her from 3pm to 1am and let her have the morning off if we needed her in the afternoon afternoon (usually for us covering mornings is easier than having somebody be home in the afternoon).
It happens maybe once or twice a year that we schedule like that (maximum) and we try to offset a late shift (midnight or later) with as much time off the following day as we can arrange (eg. we handle mornings or AP does morning but then only a few hours in the afternoon and is off at 4.30pm when I come home instead of after dinner).

The whole thing is of course much easier if/when the kids are in school. It's both a problem if you need somebody to cover days because you have an infant/toddler or during summerbreak or on snow days etc. When our children were little and home all day we chose AP to cover daytime and then got a babysitter for the evening. I know most APs will say they don't mind if they are scheduled an overnight shift in addition to their usual hours, especially not if they get paid extra, but we really don't, unless it's an emergency (and that would have to include fire, blood, broken bones or unexpectedly going into labour, not a babysitter cancelling or a last minute dinner invitation). We try to avoid having an AP that felt she couldn't say no to a request (either for the money that's in it or our of politeness or for cultural reasons or because she doesn't want to hurt our feelings). If they argue that they don't think we need to hire a babysitter because they want to pick up the extra hours for extra pay we point them to the program regulations and tell them we are sorry but they can't (we might throw in a $20 gift card to her favorite restaurant or Starbucks for a rockstar AP we disappoint by hiring a babysitter though).

But it is still a 20 hour continuous stretch. The 10 hour rule is to prevent APs from work 11, 12, 13 hour, etc continuous stretch.