Bring a stockpot of water to a boil.

Fill a large bowl with ice water and add a dash of salt. Add squash slices to boiling water and cook for 1 minute, just long enough to make the squash pliable. Immediately transfer squash to the ice bath.

Heat oil in a stockpot over medium-high heat. Add onions and garlic and sauté for 7 minutes, until onions are softened and translucent.

Add tomatoes, salt and oregano. Bring to a boil, then reduce heat to medium-low. Simmer, uncovered, for 1 hour, stirring occasionally. Add 1 tablespoon of butter, then remove from the heat.

Melt the remaining tablespoon of butter in a saucepan over medium heat. Add shallots and sauté for 4 minutes, until translucent.

Add cognac and cook for 2 minutes, until the alcohol burns off. Add mushrooms and a pinch of salt and cook for 3 minutes, until slightly softened. Transfer mushrooms to a paper towel–lined plate to drain.

In a bowl, combine three-quarters of the Parmesan with parsley, ricotta and eggs. Season generously with salt and pepper.

Preheat oven to 350°F.

Ladle a couple scoops of sauce into the bottom of a lasagna pan. Arrange a layer of squash noodles, then cheese mixture, mushrooms and sauce, alternating layers to the top. Finish with a healthy layer of sauce. Scatter the top with the remaining Parmesan.

Wrap in aluminum foil and bake for 30 minutes. Remove foil, then bake for another 10 minutes. Set aside for 15 minutes. Cut into slices, then garnish with basil. Serve hot.