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 1/2 cup cheese; repeat with two more layers (refrigerate 1/2 cup cheese for topping).
Cover slow cooker, and cook on low, 4 to 6 hours. Sprinkle lasagna with 1/2 cup cheese. Cover until cheese has melted, about 10 minutes.
To fit lasagna noodles into a round slow cooker, break off corners as needed. You don't have to precook the noodles. When the lasagna's done, they'll be ready, too.