Shuck the oysters and discard the shells. In a small bowl, stir together the buttermilk, salt and Tabasco. Marinate the oysters in the buttermilk bath for about 10 minutes.
Meanwhile, in a medium bowl, whisk together the flour, cornstarch, cayenne pepper, salt and pepper. Preheat the oil in your deep fryer or deep sided sauté pan to 350° F.
Remove each oyster one at a time and let the excess buttermilk drip off, then dredge in the flour mixture. Place in the hot oil, and working in batches, let the oysters fry for about 1 minute per side, until they are golden brown. Using a slotted spoon, remove the oysters and set them on a plate lined with paper towels to drain and cool slightly.
Make the Caper Aioli:
Peel and mash the garlic along with the salt, finely mincing and pressing with the side of the knife to create a paste. Place the garlic in a small bowl, season with some freshly cracked black pepper and whisk in the egg yolk until it is creamy. As you whisk, begin to add the olive oil gradually: begin with just a few drops of olive oil, fully whisking until the yolks thicken and emulsify. Continue adding the olive oil in drops, then in a very slow stream, continually whisking. It should be thick and very creamy. Stir in the capers, cilantro, and caper juice, and season with cayenne pepper, salt and pepper, to taste. Store in a tightly sealed container in the refrigerator until ready to serve. This can be done a few days in advance.
Buttermilk Fried Oysters inspired by and lightly adapted from Boulevard: The Cookbook (Oakes, Mazzola & Weiss, 2005). Aioli sauce adapted from The Creamiest Aioli Recipe, Suzanne Goin, via Bon Appetit (April, 2012).