Meanwhile, in a skillet over medium heat, warm 1 tablespoon avocado oil. Add the shredded squash and 1/2 teaspoon salt and a grinding of pepper and sauté until softened but still al dente, about 3 minutes. Remove and set aside.
In the bottom of a 13 by 9-inch baking pan, spread 1/2 cup tomato sauce. Top with a layer of 3 noodles. Spread 1/2 cup tomato sauce and 1/2 cup sage cream over the noodles. Sprinkle with a large handful of butternut squash and a small handful of the kale, season with a pinch of salt and a grinding of pepper, then top with another layer of 3 noodles. Repeat these steps, seasoning each layer with a pinch of salt and a grinding of pepper to use all the noodles, sauces, and veggies in a 4-layer lasagna, finishing with a layer of sauces. Top with cherry tomatoes, drizzle with 1 tablespoon oil, and add a pinch of salt and a grinding of pepper. Cover with parchment paper and bake for 25 minutes. Remove the parchment and bake until bubbling, about 20 minutes longer. Let cool for 5 to 10 minutes, sprinkle with salt, slice, and serve.