In a large pot or Dutch oven, heat 1 tablespoon olive oil (or butter) over medium-high heat. Add the chopped onion and sauté until translucent, about 2–3 minutes.
Add the minced garlic and sauté about 30 seconds, then add the cubed butternut squash and toss to combine with the onion and garlic.
Pour in the 1 1/2 cups lower-sodium vegetable broth and 1 1/2 cups milk. Bring to a boil, then reduce heat to medium-low, cover, and simmer until the squash is fork-tender, about 20 minutes.
While the squash simmers, bring a large pot of water to a boil. Add the 1 lb pasta and cook according to the package instructions until al dente. While the pasta cooks, shred the cheese if not pre-shredded.
When the squash is tender, remove the pot from the heat. Stir in the 2 tablespoons plain Greek yogurt, 1/2 teaspoon salt, and 1/2 teaspoon black pepper. Use an immersion blender to purée the mixture until smooth and creamy, or transfer the squash mixture in batches to a blender or food processor and purée, then return it to the pot.
Drain the pasta. Return the puréed squash mixture to low heat, add the drained pasta, and stir in the 2 cups shredded cheese in batches, stirring until the cheese melts and the pasta is evenly coated. If the sauce is too thick, add small splashes of milk or vegetable broth, one tablespoon at a time, until you reach the desired consistency. Taste and adjust seasoning with additional salt and pepper if needed.
Serve the mac and cheese garnished with fresh thyme as desired.