1. Adjust oven rack to middle position and heat oven to 375 degrees. Brush cut sides of squash with 2 tablespoons oil and season with salt and pepper. Place squash, cut side down, on rimmed baking sheet. Roast squash until just tender and tip of paring knife can easily be slipped into flesh, 40 to 45 minutes. Holding squash over bowl, use fork to scrape flesh from skin into strands; discard skin.

2. Dissolve salt and sugar in 2 cups cold water in container. Submerge shrimp in brine, cover, and refrigerate for 15 minutes. Remove shrimp from brine and pat dry with paper towels.

3. Heat 1 1/2 teaspoons oil in 10-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 4 minutes. Strain mixture through colander set over large bowl. Discard shells and reserve liquid (you should have about 1/3 cup). Wipe out skillet with paper towels.

4. 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.

5.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 over spaghetti squash passing lemon wedges separately.