Heat oven to 350°F (176°C). Line a 9-inch springform with 3-inch sides (see Note) with a sling of parchment paper, pressing it across the bottom and creasing the sides to get it to fit as best as possible. If the sides aren’t well covered, repeat with a second piece of parchment in the other direction.
Whisk egg and ricotta in a large bowl. Stir in half of the parmesan, fontina, water, 2 tablespoons of olive oil, salt, a few gratings of fresh nutmeg, lots of freshly ground black pepper, red pepper flakes, and garlic. Stir in squash, spinach, and sage or thyme, if using. Add dried noodles and stir until everything is coated.
Pour into prepared pan and press gently so everything is in as even of a layer as possible. Sprinkle with second half of parmesan. Gently fold any parchment that extends over the rim of the pan into the center and cover the pan tightly with foil. Bake on a sheet (for extra security against drips) for 1 hour, then remove foil, reopen the parchment folded over the top, and drizzle the dish with remaining 1 tablespoon olive oil. Return to the oven uncovered for 30 minutes. Pasta will be baked through and the top will be crisp. If it doesn’t have as much color as you’d like on top, you can finish it under the broiler for a minute or two.
Cool in pan on a rack for 30 minutes before removing the springform ring, sliding the pasta bake by its parchment onto a serving plate, and cut it into wedges.