Heat a large skillet over medium-high heat. Cook and stir beef, salt, and black pepper in the hot skillet until browned and crumbly, 5 to 7 minutes; drain and discard grease. Add onion, green bell pepper, carrot, and garlic to ground beef; cook and stir until onion is lightly browned, about 15 minutes.
Mix tomatoes, tomato sauce, mushrooms, brown sugar, and Worcestershire sauce into ground beef-vegetable mixture; season with salt and black pepper. Cook on low until liquid is reduced and flavors have blended, about 45 minutes.
Preheat oven to 350 degrees F (175 degrees C).
Bring a large pot of lightly salted water to a boil. Cook the bow-tie pasta at a boil, stirring occasionally, until cooked through yet firm to the bite, about 12 minutes; drain and transfer pasta to a 9x13-inch casserole dish.
Spoon cream of mushroom soup over bow-tie pasta; top with ground beef-tomato mixture. Sprinkle Cheddar cheese on top.
Bake in the preheated oven until casserole is bubbling and Cheddar cheese is melted, about 45 minutes.