How do I properly patch a 3/4" hole in a kitchen ceiling?
I'd use a setting-type compound (like Easysand 45) for this. Premix compound will take a long time to dry and will shrink quite a bit. Mix to a consistency of thick peanut butter. 

I'd also try to not disturb the surrounding area, filling only the hole itself as deeply as possible. Finish by dabbing compound on the patch with a small paintbrush or the corner of a rag to replicate the nearby texture. The smaller the patch, the less noticeable it will be. 

If you don't have touchup paint on hand, flat white from a spray can may work. Test on a scrap of cardboard to check the match.