Place a Dutch oven over medium-high heat and add 1 tablespoon olive oil. Let the oil heat until it shimmers, about 1 minute.

Add the ground turkey to the hot oil, breaking it into large chunks with a wooden spoon. Let it cook undisturbed for 60–90 seconds before stirring to develop browning. Continue cooking for 5–6 minutes until the turkey is browned and only slightly pink remains. Add the diced green pepper, diced onion, and a generous pinch of salt and black pepper. Stir and cook for 3–4 minutes until the onions are translucent.

Add the pressed garlic and stir for 30 seconds until fragrant. Stir in the ketchup, brown sugar, chili powder, dry mustard, paprika, cayenne pepper, and red pepper flakes. Reduce heat to medium-low and simmer for 5 minutes, stirring occasionally. Add the Worcestershire sauce and Tabasco sauce, stir to combine, and continue simmering for 15 minutes, stirring every few minutes, until the sauce has thickened and clings to the meat. If the mixture becomes too thick, add 2–4 tablespoons of water to loosen.

Check the filling has reached an internal temperature of 165°F (74°C) with an instant-read thermometer. Toast the hamburger buns under a grill or in a toaster until golden. Spoon a generous portion of the turkey filling onto each bottom bun, top with shredded sharp cheddar, and close with the top bun. Serve immediately.