Step 1: Melt the butter in a large pot or Dutch oven over medium heat. Add the sage leaves and fry for 30 seconds until crispy. Remove the sage and set aside, leaving the butter in the pot.
Step 2: Add the diced shallot to the pot and sauté for 2-3 minutes until translucent. Stir in the minced garlic and cook for another minute until fragrant, being careful not to brown it.
Step 3: Add the fresh thyme, salt, and black pepper to the pot. Stir in the diced butternut squash, cooking for about 5 minutes to allow it to slightly caramelize.
Step 4: Pour in the vegetable stock and milk. Bring the mixture to a boil, then reduce the heat to low, cover, and simmer for 10-12 minutes, or until the butternut squash is fork-tender.
Step 5: While the squash simmers, cook the pasta according to package directions in a separate pot of salted boiling water until al-dente. Reserve 1-2 cups of the starchy pasta water before draining.
Step 6: Once the squash is tender, remove the pot from the heat. Using an immersion blender, carefully blend the squash mixture until completely smooth and creamy.
Step 7: Stir in the shredded cheddar and mozzarella cheeses until fully melted and the sauce is smooth. If the sauce is too thick, add a splash of the reserved pasta water to reach your desired consistency.
Step 8: Add the cooked and drained pasta to the butternut squash cheese sauce, tossing gently until every noodle is evenly coated.
Step 9: Serve immediately, garnished with the crispy fried sage leaves and an extra sprinkle of cheese, if desired.