For a vegetarian meal, omit the sausages and substitute vegetable broth for the chicken broth. For a vegan meal, make the above changes and substitute extra-virgin olive oil for the butter.
Melt 2 tablespoons butter in 12-inch nonstick skillet over medium-high heat. Add sausage and cook, stirring occasionally, until well browned, 5 to 7 minutes. Transfer to plate and tent with foil.
Add remaining 4 tablespoons butter to fat remaining in skillet and melt over medium heat. Add carrots, tomato paste, garlic, salt, and pepper flakes and cook until beginning to caramelize, about 10 minutes. Add broth and simmer until carrots are tender and liquid is thickened, about 10 minutes.
Meanwhile, bring 4 quarts water to boil in large pot. Add pasta and 1 tablespoon salt and cook, stirring often, until al dente. Reserve 2 cups cooking water, then drain pasta and return it to pot. Transfer carrot mixture to blender. Add 1 cup reserved cooking water and blend until smooth, about 1 minute. Add sauce, sausage, and basil to pasta and stir to combine, adjusting consistency with remaining reserved cooking water as needed. Sprinkle with extra basil and serve.
