Step 1

In a large frying pan, melt the butter with 1 tablespoon of the oil over high heat. Add the mushrooms, 1/2 teaspoon of the salt, the pepper, and the dried tarragon, if using, and cook, stirring, for 3 minutes. Reduce the heat to moderately high and continue cooking, stirring occasionally, until the mushrooms are golden brown and no liquid remains in the pan, about 5 minutes.

Step 2

In a small bowl, combine the goat cheese, Parmesan, milk, and 1/4 teaspoon of the salt. Stir until smooth.

Step 3

Meanwhile, in a large pot of boiling, salted water, cook the fettuccine until just done, about 12 minutes. Reserve 1 1/2 cups of the pasta-cooking water and drain. Whisk 1 cup of the pasta-cooking water into the goat-cheese mixture. Toss the pasta with the mushrooms, the goat-cheese sauce, the remaining 1/4 teaspoon salt and 1 tablespoon olive oil, the fresh tarragon, if using, and the chives. If the pasta seems dry, add more of the reserved pasta-cooking water. Serve with additional Parmesan.