In a large stock pot, bring salted water to a boil. Debeard mussels. Set aside.
In a large saute pan, over high heat, combine the butter and olive oil. Add the shallots and garlic and cook for 1 to 2 minutes until translucent. Do not brown. Add the chili flakes and stir to combine.
Add the reserved mussels, clams, tarragon, thyme, and rosemary and continue to saute until herbs begin to turn fragrant. Deglaze with white wine and season with salt and pepper. Cover the pan and continue to cook over high heat until the shellfish have opened, about 3 to 4 minutes.
Meanwhile, add a couple tablespoons of salt and a drizzle of olive oil to the boiling water. Add the pasta and cook until al dente. Drain. Remove cover from saucepan containing the shellfish and discard the sprig of rosemary. Taste the sauce and adjust seasoning, if necessary. Add the cooked pasta, chopped parsley and tarragon, and Parmesan. Toss with sauce and shellfish until well blended. Serve immediately.