Preheat the oven to 375°F.
Heat the oil in a large Dutch oven or saucepan over medium-high heat. Add the garlic, onion, mushrooms, fresh and dried basil, and salt and saute until the vegetables are softened, 4 minutes. Add the sausage and ground turkey, breaking it up with a wooden spoon and stirring to combine. Cook the mixture until the meats are opaque and cooked through, about 12 minutes, stirring often.
Place the tomatoes in a blender and puree until completely smooth. Add the tomato puree and tomato paste to the Dutch oven and cook until thickened slightly, 5 minutes. Set aside to let cool slightly.
Meanwhile bring a large pot of water to a boil, add the pasta, and cook to your preferred doneness, stirring occasionally. Drain in a colander.
Assemble the lasagna in a deep 9 x 13-inch casserole dish. Begin with a layer of noodles, using about 4 noodles to cover the bottom of the dish. Spread one quarter of the cottage cheese over the noodles, followed by one quarter of the sauce. Sprinkle 1 cup each of the Monterey Jack and white Cheddar cheeses. Repeat to make 4 more layers, for 4 total layers.
Cover and bake for 30 minutes. Uncover and bake 15 minutes more, or until the cheese is melted and bubbly. Let cool slightly to firm up, garnish with the basil, then slice and serve.