Preparation Bring 6 cups water with 1 teaspoon salt to a simmer in a wide 6- to 8-quart pot, then add oysters with their liquor and poach at a bare simmer just until edges curl, about 1 1/2 minutes. Transfer oysters with about 1/4 cup poaching liquid to a bowl and chill, uncovered, until cool, about 30 minutes. Scrub reserved bottom oyster shells. Boil shells in 6- to 8-quart pot of boiling water 3 minutes (to remove any grit or traces of raw oyster), then drain in a colander. Rinse shells and dry thoroughly. Stir together vinegar, apple juice, shallot, beet, peppercorns, salt, sugar, and 2 tablespoons poaching liquid from bowl of oysters. Very thinly julienne cucumber lengthwise using a mandoline or other manual slicer, working around seed core, then discard seed core. Drain cucumber in a sieve set over a bowl while assembling oysters, then discard liquid. To assemble, place 1 oyster in each half shell and spoon 1 teaspoon mignonnette over it. Top with a little cucumber, twirled into a mound. Serve immediately.