Place an oven rack about 6 inches below the broiler and preheat the broiler. Place the peppers on a broiler pan or baking sheet and broil, turning occasionally, until the skin is blackened and blistered, 12-15 minutes. Remove the peppers from the oven and place in bowl. Cover the bowl with plastic wrap, and let the peppers sit until cool, 10 to 15 minutes. Use a small knife or your hands to remove the skin, then remove the seeds and stem from the pepper.

Place the peppers in a food processor along with the olive oil, Parmigiano-Reggiano, basil, half and half and garlic. Process until smooth. Taste and season with salt and pepper to taste. At this point, you can cover the pesto and let it sit at room temperature for up to 2 hours.

Bring a large pot of water to a boil. Add the linguine and cook until al dente. Reserve a mug full of the starchy cooking water before draining the pasta. Once drained, return the pasta to the pan. Add about 3/4 of the pesto and as much of the cooking water as needed to make a creamy sauce.

Transfer the pasta to a serving plate and top with the remaining pesto. Serve with additional Parmigiano-Reggiano.