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.
2. In a small bowl, combine the goat cheese, Parmesan, milk, and 1/4 teaspoon of the salt. Stir until smooth.
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.
Variation: Use fresh basil instead of or in addition to the tarragon and chives in the recipe.
Wine Recommendation: Dolcetto, while beloved in Piedmont, is less appreciated in the United States. Yet its light tannins, cherry fruit, and clean, dry finish are just the ticket to balance the full flavor of the cheeses here.