In a medium pot, melt the butter over medium low heat.
Once melted, add the sliced onion and cook, stirring occasionally, until deeply golden, about 20 minutes.
Once golden, add the garlic and cook just until fragrant, 30 seconds to 1 minute. Transfer the onions and garlic to a blender.
Pour the vegetable broth into the pot and add the cubed butternut squash.
Cook over medium heat until the squash is completely tender, 10-15 minutes.
Transfer 1/2 cup (120 milliliters) of the vegetable broth to the blender.
Drain the squash and add to the blender.
While the squash is cooking, bring another large pot of salted water to a boil. Add the pasta and cook until just tender, al dente. Drain and set aside.
Add the milk to the blender with the squash and process until the mixture becomes a smooth, thick sauce.
Pour the sauce back into the medium pot and stir in the shredded Cheddar and Parmesan cheese until melted.
Stir in the Dijon mustard, salt, paprika, and black pepper. Adjust flavors to taste.
Fold the sauce into the drained pasta and serve immediately with parsley and more Parmesan for garnish.