Heat 1 tablespoon olive oil in a large skillet over medium-high heat. Add the ground chicken or turkey and cook, breaking up with a spatula, until browned, about 5–6 minutes.
Add the sliced onion, sliced green pepper, and sliced mushrooms (if using) to the skillet. Cook until the vegetables are tender and onions are starting to soften, about 5–10 minutes.
Stir in 2 tablespoons Worcestershire sauce, 1/4 cup soy sauce, and 1 tablespoon ketchup. Season with 1/4 teaspoon salt and 1/4 teaspoon black pepper. Bring to a simmer and cook about 2 minutes, then remove from heat.
While the chicken mixture cooks, make the cheese sauce: melt 1/2 tablespoon butter in a small saucepan over medium heat. Whisk in 1/2 tablespoon flour and cook for about 1 minute.
Slowly whisk in 3/4 cup milk and 1/4 teaspoon mustard powder. Bring to a gentle bubble and cook until slightly thickened, about 2 minutes.
Turn off the heat and stir in 1 1/2 cups shredded provolone and 2 tablespoons grated Parmesan until melted. Season with a pinch of salt and pepper to taste.
Toast the slider buns (or regular buns) if desired. Spoon the chicken and pepper mixture onto the bottom halves, top with the cheese sauce, and place the bun tops on to serve.
