Step 1: Liquid Nitrogen can freeze alcohol
Geek note: Most drinks are not 200 proof - so the water-alcohol mixtures freeze at a higher temperature than pure alcohol. Still, it's pretty darn cold.
Step 2: Liquid Nitrogen Safety
Since we're freezing alcohol to -114 C, it's important to realize the frozen cocktail itself can be temperature-wise dangerous. At our party, we froze the cocktails, and carefully warned anyone who wanted to try the cocktail-ice directly. In most of the drinks, we also added liquid to some of the cocktail-ice to prevent people from coming into direct contact with the super-cold solids. And in the ones where we didn't, it was probably a bad idea.
I've always gotten my LN2 from a university, or by masquerading as a "remote outpost" of a university to that university's supplier. Renting the dewars isn't too expensive, and the cost of the LN2 itself is less then milk per gallon.
Step 3: Materials, tools, and setup
You can use a wooden spoon and a bowl with a handle, but it's a lot of work, the bowl gets very cold, and the LN2 and the mix may splash onto your hands.
Better to lock the bowl to the table and use a power stirrer.
We use the bowl from my 4.5 quart KItchenAid mixer, with a new "lock plate".
(I got mine online at Goodmans.net )
The stirring bit is a spackle mud stirrer (these are heavier duty than the paint stirrers available at my hardware store) chucked into my cordless drill.
Note that this setup put little dings into the mixer bowl, which are unsightly and, my wife assures me, will require the acquisition of a new bowl for the mixer.
Step 4: Mix it up!
Step 5: Frozen Mimosa
Traditionally equal parts OJ and champagne, but
the low temp suppress the flavor, so we went about 2 parts OJ to 1 champagne.
Put the frozen mix into a glass, add a splash of liquid champagne and a maraschino cherry!
Step 6: Frozen Orange Juice and Frozen Apple Juice
Step 7: Frozen Cosmos
Mix 2 parts lemon vodka to one part triple sec combined with one part cranberry and the juice of half a lime. A splash of cranberry afterwards serves to convert the ice to more of a slushie and avoids oral liquid nitrogen burns -- never fun.