Preheat oven to 400 degrees F.
Cook 2 ¼ cups pasta according to package directions. Drain; set aside.
Meanwhile, heat 1 tablespoon oil in a 12-inch cast-iron skillet over high heat.
Add 1 pound turkey; cook, stirring to crumble, until lightly browned, 4 to 5 minutes.
Add 2 ½ cups cauliflower, 2 cups bell peppers, 1 ½ cups onions, 1 cup carrots and 5 minced cloves garlic; cook, stirring often, until the vegetables are softened, 5 to 7 minutes.
Add 1 can tomato sauce, 1 ½ tablespoons brown sugar, 1 tablespoon Worcestershire, 2 teaspoons mustard, 1 ¼ teaspoons salt and 1 teaspoon pepper.
Bring to a boil over medium-high heat. Reduce heat to medium; simmer, stirring occasionally, for 2 minutes.
Stir the cooked pasta into the turkey mixture; sprinkle evenly with 1 cup cheese.
Bake until the cheese is melted, 5 to 7 minutes.
