BBQ chicken . . . indoors? No, we’re not pulling your leg! (Get it?)
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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.
Use clean tongs to transfer chicken to large plate. Let cool slightly, then use 2 forks to shred chicken into bite-size pieces (see photo, below). Return shredded chicken to skillet and stir to coat with sauce.
Start by making a sweet and tangy barbecue sauce in a skillet and then cook the chicken right in the sauce, which helps it absorb lots of great flavor. Two forks make quick work of shredding (or “pulling”) the cooked chicken (it’s even quicker if you ask a sibling or friend to help!).
Cutting the chicken breast in half lengthwise before cooking it in the sauce, makes it easier to shred the meat into bite-size pieces after it is cooked. Here's how to do it: