1. Arrange a rack in the middle of the oven and heat to 425°F.
2. Halve the squash lengthwise and scoop out the seeds. Transfer to a sheet pan. Drizzle with 1 tablespoon of the oil and season with ½ teaspoon of the salt and several grinds black pepper. Add the garlic and toss to coat in any residual oil. Flip the squash cut side down and arrange the garlic in the cavities to prevent burning. Roast until tender, 40 to 45 minutes.
3. Meanwhile, combine the Panko, ½ cup of the Parmesan, remaining 1 tablespoon olive oil, and a pinch each of salt and pepper in a medium bowl. Set aside.
4. Bring a large pot of salted water to a boil. Cook the pasta according to package directions until al dente. Reserve 1 cup of pasta water and drain.
5. When cool enough to handle, scoop out the flesh of the squash and squeeze the garlic out of their peels and add to the bowl of a food processor. Add the cottage cheese, nutmeg, cayenne, remaining ½ cup Parmesan, and remaining 1 teaspoon salt. Process until smooth, scraping down the sides as needed. With the motor running, slowly add the broth until the sauce is thick yet pourable (you may not need the full amount). Taste and season with salt and pepper.
6. Return the pasta to the pot, add the sauce, and toss, adding a few splashes of pasta water until the sauce evenly coats the pasta. Heat the broiler.
7. Transfer the pasta to the now-empty sheet pan and spread into an even layer. Top with the bread crumb mixture. Broil until the bread crumbs are golden brown, 2 to 4 minutes. Garnish with fresh thyme leaves.