Prep the squash. Heat the oven to 400 degrees F. Cut the squash in half lengthwise and scoop out the seeds. If it is hard to cut, pierce it a few times with a fork and microwave for 2 to 3 minutes first. Rub the cut sides with 1 tablespoon of the oil, then season with the salt, pepper, smoked paprika, and nutmeg.

Roast. Put the squash cut side down on a parchment-lined sheet. Drizzle the garlic head with the remaining oil, wrap it in foil, and set it on the same pan. Roast for 40 to 45 minutes, until a fork slides easily into the squash.

Cook the pasta. Boil the pasta in well-salted water until 1 minute short of al dente. Save 1 cup of the pasta water, then drain.

Blend. Scoop about 2 cups of the squash flesh into a blender and add 4 to 6 squeezed-out roasted garlic cloves, the cottage cheese, milk, 1/4 cup pasta water, Dijon, lemon, and salt. Blend until completely smooth, about 60 seconds. Add more milk or pasta water a splash at a time until it is the texture of a thick soup.

Melt in the cheese. Pour the sauce into a large pot over low heat. Add the cheddar and Gruyere a handful at a time, stirring until melted. Keep the heat low so the cottage cheese does not turn grainy. Taste and add salt, and more Dijon or lemon if it tastes flat.

Combine. Add the pasta and stir to coat, loosening with more pasta water until it is glossy and creamy. Serve right away.

Optional bake. Add an extra splash of milk, since it thickens in the oven. Transfer to a baking dish, top with extra cheese and the buttered panko, and bake at 375 degrees F for 15 to 20 minutes, until golden.