Heat 1 tablespoon olive oil in a large skillet over medium-high heat until shimmering.
Add 1 pound ground chicken and 1/4 teaspoon kosher salt to the skillet and cook, breaking up the meat, until browned and no longer pink, about 5–7 minutes.
Stir in the diced 1/2 onion, diced 1/2 green bell pepper, and 2 cloves minced garlic; cook until the vegetables are softened, about 2–3 minutes.
Add 1 teaspoon chili powder, 1/2 cup ketchup, 1/2 tablespoon mustard, 1/4 cup tomato paste, 1 teaspoon Worcestershire sauce, and 1/2 cup water; stir to combine.
Reduce heat to low, cover, and simmer for 10 minutes, stirring occasionally; uncover and simmer a few minutes more if you want it thicker, or add a little water if it’s too thick.
Adjust seasoning with salt to taste, then spoon the mixture onto the 4 hamburger buns and serve.