In a 5-quart Dutch oven or heavy pot, cook sausage and beef over medium-high, breaking up meat with a spoon, until no longer pink, 4 to 6 minutes.
Add onion, carrots, and garlic; season with salt and pepper. Cook until onion has softened, 3 to 5 minutes.
Stir in tomato paste, then tomatoes; bring to a boil, and remove from heat.
Spoon 2 cups meat mixture into bottom of a 5- to 6-quart slow cooker. Layer 3 noodles (breaking them as needed to fit), 2 cups meat mixture, and ½ cup cheese; repeat with two more layers (refrigerate ½ cup cheese for topping).
Cover slow cooker, and cook on low, 4 to 6 hours.
Sprinkle lasagna with ½ cup cheese. Cover until cheese has melted, about 10 minutes.
