1. 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, 4-6 minutes. Add onion, carrots, and garlic; season with salt and pepper. Cook until onion has softened, 3-5 minutes. Stir in tomato paste, then tomatoes; bring to a boil, and remove from heat.
2. Spoon 2 cups meat mixture into bottom of 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).
3. Cover slow cooker, and cook on low for 4 to 6 hours. Add some beef broth if it looks dry and let cook a little longer. Sprinkle lasagna with 1/2 cup cheese. Cover until cheese has melted, about 10 minutes.
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