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.