Remove the stem and seeds from the Hatch chiles and discard. Slice the chiles into thin strips and set aside.
Bring a large pot of salted water to a boil. Once boiling, add the fettucine. Cook until al dente, about 8 minutes. Reserve 1/2 cup of the cooking liquid and then strain the water from the pasta. Keep the cooked pasta in the pan it was cooked in. Set aside.
Meanwhile, add the butter into a large skillet. Melt the butter over medium heat. Add the shallot and Hatch chiles to the skillet and cook for 3 minutes. Add the garlic and cook until fragrant, about 30 seconds.
Decrease the heat to medium-low and add the cream, simmer for 5 minutes.
Add the reserved cooking liquid, parmesan, salt and pasta into the sauce. Stir and toss to thoroughly coat and combine. Cook until the sauce is thickened and the pasta warmed through, about 2 to 3 minutes.
Place in warm serving bowls and serve immediately topped with parmesan.