BBQ chicken . . . indoors? No, we’re not pulling your leg! (Get it?)
In liquid measuring cup, whisk together ketchup, molasses, mustard, vinegar, and Worcestershire sauce. Set aside.
In 12-inch skillet, heat oil over medium heat for 1 minute (oil should be hot but not smoking). Add shallot, chili powder, and salt. Cook, stirring occasionally with rubber spatula, until shallot is softened, about 3 minutes. Stir in ketchup mixture, scraping up any browned bits.
Use tongs to carefully place chicken in skillet. Bring mixture to boil. Reduce heat to medium-low, cover, and simmer for 4 minutes.
Use oven mitts to remove lid. Use clean tongs to flip chicken pieces over. Cover and simmer until chicken registers 165 degrees on instant-read thermometer, 4 to 6 minutes. Turn off heat. Slide skillet to cool burner.