In a skillet over medium heat, warm 2 tablespoons olive oil until it shimmers. Add squash and a generous pinch of salt and pepper. Cook until squash takes on some color and starts to soften, about 8 minutes. Add garlic and red pepper flakes and cook 1 minute. Add broth, bring to a simmer; cover and cook until squash is tender, 7–8 minutes. Cool slightly, then use an immersion blender to purée smooth. Taste and add salt and pepper as needed.

Season shrimp with salt and pepper. In a skillet over medium heat, warm 2 tablespoons olive oil. Add shrimp, sauté 3 minutes; turn and cook another 1–2 minutes, just until shrimp are pink.

Drain the pasta, reserving ½ cup of cooking water. Return to pot set over low heat. Add squash purée and milk, stirring to create a glossy sauce that coats the pasta. Add a splash of cooking water if needed. Add Parmesan and stir until incorporated. Plate pasta; top with shrimp, basil and additional Parmesan.