Bring a large pot of water to a boil and cook pasta according to package directions.
While pasta is cooking, preheat a deep skillet over medium-high heat.
When the skillet is hot, add butter and swirl to coat the bottom. Add garlic and cook, until fragrant, about 1 minute.
Add beans, milk, salt, and pepper to the skillet. Bring to a simmer, then lower heat and continue to cook, stirring often until beans are tender, 5-7 minutes.
Use a blender, food processor, or hand blender to blend mixture into a smooth sauce.
Return sauce to the skillet over medium heat. Add Parmesan cheese and continue to cook, stirring often, until cheese is melted and sauce thickens slightly, about 10 minutes. Sauce will also thicken a bit more as it sits. Taste and adjust seasonings, if necessary.
Serve sauce over pasta and top with more pepper and Parmesan, if desired.