I once attended a wedding in Umbria during truffle season. They put truffles on a lot of stuff - scrambled eggs, pasta, cheese on toast etc. If you wanted something more labor intensive you could make a mushroom tart or something and top that with the truffles.
I wouldn’t combine truffle oil with real truffles in oil.

I guess I use truffles more for Italian? I'd put it over really good mushroom ravioli. Also I like toasting a piece of great bread (I know I sound like Ina here...) and melting some Buffalo mozzarella or burrata on top and then dousing in the truffle oil and slivers. Super simple but really delicious, assuming you love truffles as they're the star.

Over an omelette, bien sur.

I brought home from Paris last spring, a jar of slivered truffle carpaccio. Please give me some ideas on how to go about using it to its full potential. I’m an excellent cook, but new to French cooking and don’t know how to use it intuitively.