Bring a large pot of water to a boil. Add the pasta, and cook according to package directions, about 9 minutes. Drain, and set aside.
In a large stockpot over medium heat, add 2 tablespoons of butter. Then, add the garlic, butternut squash, and vegetable broth. Bring to a boil, and lower to a simmer for 5 minutes.
Using an immersion blender, blend the squash and broth mixture until very smooth.
Add the milk, remaining 2 tablespoons of butter, cheddar cheese, salt, and pepper. Stir over medium heat until the cheese melts, about 5 minutes.
Add the drained pasta to the cheese sauce. Stir well.