Bring a large stock pot of salted water to a rolling boil. Add the linguine and cook according to the package directions until al dente. Reserve 1 cup of the cooking liquid then drain.
While the pasta cooks, make the sauce: Heat a 12-inch nonstick skillet over medium heat. Add the butter and cook until bubbly, browned and smells nutty. Reduce the heat to low; add the creme fraiche, the zest from both lemons, 1 teaspoon salt and 1/2 teaspoon pepper. Stir together until a smooth, creamy sauce forms.
Add the crabmeat and stir gently. Once the pasta has cooked, add it directly to the sauce
along with the lemon juice. Toss together then stir in the 1/4 cup pecorino, minced dill, 2 tablespoons chives and the tarragon. Cook until the sauce thickens, about 1 minute. If it looks too dry, stir in 1/4 cup of the reserved cooking liquid at a time. Remove from the heat and taste for seasoning, adding more salt or pepper if needed.
Divide the pasta among 4 shallow bowls and sprinkle the tops with more pecorino, cracked pepper, minced chives and dill fronds.