Dissolve salt and sugar in 1 quart cold water in large container. Submerge shrimp in brine, cover, and refrigerate for 15 minutes. Remove shrimp from brine and pat dry with paper towels.

Heat 1 tablespoon oil in 12-inch skillet over high heat until shimmering. Add shrimp shells and cook, stirring frequently, until they begin to turn spotty brown and skillet starts to brown, 2 to 4 minutes. Remove skillet from heat and carefully add wine and thyme sprigs. When bubbling subsides, return skillet to medium heat and simmer gently, stirring occasionally, for 5 minutes. Strain mixture through colander set over large bowl. Discard shells and reserve liquid (you should have about ⅔ cup). Wipe out skillet with paper towels.

Combine lemon juice and cornstarch in small bowl. Heat remaining 1 tablespoon oil, garlic, pepper flakes, and pepper in now-empty skillet over medium-low heat, stirring occasionally, until garlic is fragrant and just beginning to brown at edges, 3 to 5 minutes. Add reserved wine mixture, increase heat to high, and bring to simmer. Reduce heat to medium, add shrimp, cover, and cook, stirring occasionally, until shrimp are just opaque, 5 to 7 minutes. Remove skillet from heat and, using slotted spoon, transfer shrimp to bowl.

Return skillet to medium heat, add lemon juice–cornstarch mixture, and cook until slightly thickened, 1 minute. Remove from heat and whisk in butter and parsley until combined. Return shrimp and any accumulated juices to skillet and toss to combine. Serve, passing lemon wedges separately.