Dissolve the 3 tablespoons salt and 2 tablespoons sugar in 1 quart of cold water in a large gallon-size freezer ziplock bag. Add the 1 ½ pounds of shrimp to the brine and refrigerate for 15 minutes (no longer or the shrimp will turn mushy). Remove shrimp from brine and pat dry with paper towels.
Heat 1 tablespoon oil in a 12-inch skillet over high heat until shimmering. Add the shrimp shells and cook, stirring frequently, until they begin to turn spotty brown, 2 to 4 minutes. Remove skillet from heat and carefully add the 1 ½ cup of wine and the 5 thyme sprigs. When bubbling subsides, turn the heat to medium and return the skillet to stove. Simmer gently, stirring occasionally, for 5 minutes. Strain mixture through a sieve over a large bowl. Discard shells and reserve the liquid (you should have about ⅔ cup). Wipe out skillet with paper towels.
Combine 3 tablespoons lemon juice and 1 teaspoon cornstarch in a small bowl. Set aside. Heat remaining 1 tablespoon oil, sliced garlic, red pepper flakes, and black pepper in the now-empty skillet over medium-low heat, stirring occasionally, until the garlic is fragrant and just beginning to turn golden at edges, 3 to 4 minutes. Add the reserved wine mixture, increase heat to high, and bring to a simmer. Reduce heat to medium now, add the shrimp, cover, and cook, stirring occasionally, until the shrimp are just opaque, not yet fully cooked, 5 to 7 minutes. Remove skillet from heat and using a slotted spoon, transfer shrimp to a bowl.
Return the skillet to medium heat, add the lemon juice-cornstarch mixture, and cook until slightly thickened, this should happen very quickly. Remove from heat and whisk in the 4 tablespoons of butter and 1 tablespoon of chopped parsley until butter has melted. Return shrimp and any accumulated juices to the skillet and toss to combine. The shrimp should be fully cooked but not over-cooked now, between resting in the bowl and being added back to the sauce mixture.