Heat a large skillet over medium heat. Place the 4 split buns cut-side down in the dry skillet and toast for 1–2 minutes. Transfer them to four plates.
Add 1 tbsp olive oil, 1 diced half onion, and 1 diced green bell pepper to the skillet. Cook for 2 minutes.
Add the 1 lb ground chicken, 1/2 tsp kosher salt, and 1/4 tsp black pepper. Cook for 7–9 minutes, breaking up the chicken, until a food thermometer inserted into the center reads 165°F.
Stir in the 3/4 cup barbecue sauce and 1 tbsp Worcestershire sauce. Simmer for 2 minutes.
Lay the 4 cheddar slices over the chicken mixture in the skillet. Cover and cook for 1 minute, until the cheese melts. Divide the cheesy chicken mixture among the bottom buns and close the sandwiches with the top buns.
