Warm 3 tablespoons of olive oil in a large, heavy skillet over medium-high heat. Add the butternut squash, garlic, 1 teaspoon salt, and 1/4 teaspoon freshly ground black pepper. Saute until the squash is golden and tender, 5 to 7 minutes. Add the vegetable stock, bring to a simmer, cover and cook until the squash is very soft, another 5 to 7 minutes. Transfer the squash mixture to a blender or food processor and puree.
Bring a large pot of salted water to a boil over high heat. Add the pasta and cook until tender but still firm to the bite, stirring occasionally, about 8 to 10 minutes. Drain pasta.
Meanwhile, warm the remaining 3 tablespoons of olive oil in a large, heavy skillet over medium-high heat. Sprinkle the prawns with the remaining 1 teaspoon of salt and remaining 1/2 teaspoon of freshly ground black pepper. Add the prawns to the pan and cook until just pink, about 3 minutes.
In a large pot over low heat combine the cooked pasta, pureed squash mixture, and 3/4 cup milk. Stir to combine. Add the remaining 1/4 cup milk if the sauce needs to be moistened. Add the cooked prawns, basil, and cheese. Stir until warm and serve.