The title is the only reason I'm here! haha.
The faster you bring the water to freezing temperature, the smaller the crystals will be and the less "good stuff" that will be lost in the mix. Also, I would assume that would raise your effective "separation" (lets not use the D word, fractional or otherwise) percentage. Obviously dry ice is about as fast as it gets.
Have you made apple jack before? I've done it once and it turned out well, however it was a young batch of cider and I want to try it again with something with a bit more age/tastiness on it someday. I suppose for doing shots it won't matter as much.
I froze mine in a freezer, just in a big bowl with a mesh strainer set in it. The ice really seemed to seed/form on the strainer so every 30 min or so I'd pull the strainer out and knock the ice off and put it back in. Since the freezing water was on the top of the bowl the strainer picked that up too. Something like that might work well with the dry ice too, although with how quickly that will go maybe just letting the water go solid would be better.
I wonder what would happen if you just chucked the dry ice into the cider... hmm...
Drinking: Orfy's MMA, De Belge Farmhouse IPA, Late for Breakfast Stout
Bottle conditioning: Xmas Apfelwein, How Rye Am I? Saison
Primaries: Graff, Leftovers American Honey Wheat