In large skillet over medium high heat, heat olive oil. Add carrots and shallots, season with salt and pepper, and sauté until softened, 3 to 4 minutes.
Add garlic and red pepper flakes; stir and cook until fragrant, 1 minute.
Add broth and vinegar, bring to a boil, cover and reduce heat to medium-low. Simmer for 20 minutes or until carrots are very tender.
Meanwhile, bring a large pot of salted water to boil. Add pasta and cook according to package directions. Reserve ½ cup of pasta water; drain pasta.
When carrots are tender, carefully transfer carrot mixture to a blender and purée until smooth. Add cream and blend until smooth.
Return sauce to pan. Add cooked pasta and stir to coat. Thin with reserved pasta water, if needed.
Serve topped with Parmesan cheese, additional chives, and red pepper flakes, if desired.
Notes
Carrot sauce can be served with any type of pasta, including whole wheat, gluten-free or regular.
More Topping Ideas: Instead of Parmesan and chives, top the pasta with a spoonful of pesto or toasted bread crumbs.