If you can get ahold of pisco, a pisco sour (Peruvian) is a fun classic. https://en.m.wikipedia.org/wiki/Pisco_Sour

And generally, most of the sour family, and their descendents the fizzes. https://en.m.wikipedia.org/wiki/Sour_(cocktail)

Egg whites add a nice... balance and mouthfeel to the cocktail that you don't really get from other ingredients.

For the yolks, flips are the big family (think nog without cream). https://en.m.wikipedia.org/wiki/Flip_(cocktail)

You can also save the yolks and make zabaione without being wasteful. Heavenly decadent on fresh fruit and easy to whip up after dinner. Also a great wow if you've got leftover champagne, as the carbonation isn't particularly important. https://en.m.wikipedia.org/wiki/Zabaione

I usually split an egg and store the unused portion (whites or yolks) in the fridge 24 hours and use them the next day.

My personal favorite egg white cocktail is the Clover Club, which hasn’t been mentioned yet.