Cook the meat and carrot in a large nonstick skillet over medium-high heat, breaking up the large pieces, until no longer pink, about 5 minutes. Drain excess fat. Preheat the oven to 375°F. Add the salsa, black beans, corn, chili powder, and cumin to the skillet and stir to combine. To assemble the lasagna, arrange a third (about 2 cups) of the meat mixture in a 9 x 13-inch baking dish. Layer half the tortillas over the meat, allowing them to overlap. Spoon half of the cottage cheese and 1/2 cup of the Cheddar cheese over the tortillas and spread evenly. Place 2 more cups of meat mixture over the cottage cheese. Layer with the remaining tortillas and cottage cheese. End with the meat mixture. Top with the remaining Cheddar cheese and bake uncovered until the cheese melts and the lasagna is heated through, about 25 minutes.