In a large container with a lid, dissolve 6 tablespoons salt in 3 quarts of cold water. Cut turkey into quarters and submerge in brine. Cover and refrigerate for 30 minutes.
Adjust an oven rack to the lower-middle position and preheat oven to 325°F (165°C).
In a Dutch oven over medium heat, warm 1 tablespoon oil until shimmering. Add onion and cook until softened, about 5 minutes. Stir in garlic, chili powder, and cayenne and cook until fragrant, about 30 seconds. Whisk in ketchup, 1 tablespoon vinegar, Worcestershire, mustard, molasses, and hot sauce, and bring to a simmer.
Remove turkey from brine and pat dry with paper towels. Arrange turkey pieces in sauce and turn to coat. Cover Dutch oven, move it to the oven, and cook until turkey registers 160°F (71°C), 25 to 40 minutes.
Transfer turkey to a cutting board and let cool slightly, about 5 minutes. Using 2 forks or your fingers, shred turkey. Meanwhile, bring remaining sauce in Dutch oven to a gentle simmer over medium heat and cook, stirring often, until slightly thickened, 2 to 5 minutes.
Stir shredded turkey and any accumulated juices into sauce and cook until warmed through, about 1 minute. Season with salt and pepper to taste.