Preheat the oven to 390F / 200C and line a baking sheet with parchment paper.
Slice the peppers into 2-3 large pieces, removing the stalk, seeds and white membrane. Peel and roughly chop the carrots. Add to the baking sheet and drizzle with olive oil. Roast the peppers and carrots for 30 minutes.
Once roasted, add the peppers to a bowl and cover with a plate for 5-10 minutes. Peel the skins from the peppers once they are cool enough to handle.
While the peppers are roasting, cook the pasta to package instructions in salted water. Reserve about half a cup of the pasta water. Drain the pasta.
Drizzle olive oil in a skillet and cook the diced onion for 4-5 minutes, then add the minced garlic and sweet paprika and cook for a further minute.
Add the roasted peppers, carrots, onion mix and fresh thyme leaves to a blender and blend to a sauce-like consistency.
Pour the sauce into the skillet, and tip in the cooked pasta. Pour in the heavy cream and a couple of tablespoons of the reserved pasta water into the skillet, and stir well, adding more of the water to thicken the sauce so it coats the pasta as needed. Season with salt and black pepper, if using. Spoon into your pasta bowls and garnish with fresh basil leaves.