I think any dough is going to be challenging with that, but the sugar cookie dough I use really doesn’t rise much and is quite good. It also doesn’t need to be refrigerated before baking. I recommend using almond extract. It’s also delicious with KAF Measure for Measure flour if you ever need gluten free.
https://www.inkatrinaskitchen.com/best-sugar-cookie-recipe-ever/
A cookie with nono leavening is going to not be very tasty.