Air Fryer Juicy Lucy Mushroom Burgers

Name: Air Fryer Juicy Lucy Mushroom Burgers
Ingredients: Four 4-inch portobello mushrooms, stems removed; 1 tablespoon olive oil; Kosher salt and freshly ground black pepper; 3 ounces white Cheddar, shredded (about 1 cup); 3 ounces Havarti cheese, shredded (about 1 cup); 2 ounces cream cheese, at room temperature; 1/4 teaspoon chili powder; 1/4 teaspoon garlic powder; 1/4 teaspoon onion powder; 1 cup all-purpose flour; 1 large egg; 1 cup panko; Olive oil cooking spray; 1/4 cup mayonnaise; 2 tablespoons barbecue sauce; 1 tablespoon ketchup; 4 brioche burger buns, split; 4 leaves bibb or green leaf lettuce; 1 large tomato, sliced