Preheat the oven to 375 F and lightly grease a 9x13” baking dish with olive oil. Cook the lasagne noodles to al dente, drain and set aside. Make the other layers below while the lasagne noodles cook.
To prepare the squash layer, heat the olive oil in a large saucepan or pot over medium heat on the stove and add the onion. Cook until the onion is transparent, 3-5 minutes stirring, and add the garlic. Cook an additional 1-2 minutes until the garlic is fragrant, stirring. Add the squash, stock and salt and pepper, turn to low, and cover. Cook until the squash is soft, approximately 25 -30 minutes. Uncover and cook an additional 10 minutes, until the liquid is cooked off of the squash. Using an immersion blender, blend the squash until smooth. Season with salt and pepper to taste and set aside.
To prepare the cottage cheese layer, mix together all of the ingredients for that layer in a medium bowl and set aside.
To assemble the lasagne, layer 3 noodles in the prepared dish length-ways, and evenly spread the cottage cheese layer on top of the noodles. Sprinkle with ½ cup of the grated mozzarella cheese and top with 3 more noodles. Spread ¾ of the squash layer on top of the next noodle layer and sprinkle with the another ½ cup of grated mozzarella cheese. Top with the last layer of 3 noodles and spread the remaining squash puree over the top of the noodles. Place the mozarella slices evenly to cover the top of the dish, sprinkle with the parmesan cheese and remaining grated mozzarella cheese and evenly place the sage leaves on the top of the lasagne.
Place in the oven and bake for approximately 45 minutes (uncovered), until the lasagne bubbles and turns golden on top. Allow to cool for 15 minutes before serving.