Preheat oven to 400ºF.

Place the diced squash and thyme sprigs in an ovenproof skillet and drizzle with 1 tablespoon oil. Season all over with salt and pepper. Nestle the head of garlic into the squash, cut side down.

Transfer to the oven for 30–45 minutes, flipping once while cooking, until the squash is tender and browned. Check the garlic occasionally to ensure the tops aren't burning.

Before the squash finishes roasting, bring a large pot of water to a boil. Add salt and the pasta. Cook until al dente. Scoop out 1 cup of pasta cooking water. Set it aside and then drain the pasta.

Remove the skillet from the oven. Transfer the garlic to a plate and let it cool. Remove and discard the thyme sprigs. Once the garlic is cool enough to handle, carefully pop the cloves out from the skin. Transfer the garlic to the skillet. Using a fork or potato masher, mash squash and garlic until mostly smooth. Taste and season with salt and pepper.

Add the pasta cooking water and cream to the skillet and stir until smooth. If needed, continue to thin out the sauce with more water, cream, or vegetable broth until the sauce reaches desired consistency. Bring to a low boil, reduce heat, and simmer for 10 minutes.

Add the cooked pasta to the sauce and toss to coat. Taste and season once more. Turn off the heat.

Finish the pasta with a splash of lemon juice if you like. Divide between bowls and finish with freshly minced parsley or chives, if you like. Enjoy!