In a large bowl, mash ricotta and goat cheeses with the milk until soft. Add the nutmeg, ¼ cup of Cacio di Roma cheese, and salt and pepper, and stir well. Cover the mixture and refrigerate until firm, about 30 minutes.
Using the pasta machine, roll out the pasta to the thinnest setting. Using a biscuit cutter or water glass, cut out 2-inch circles. Pipe or carefully spoon a rounded tablespoon of the cheese filling onto the center of half of the rounds and cover the filling with a second pasta round. Press the edges together firmly to seal.
Bring 6 quarts of water to a boil and add 2 tablespoons of salt. Drop the tortelli into the boiling water, and cook until tender, 3 to 5 minutes. Carefully drain the tortelli, reserving about 1 cup of the pasta water.
Meanwhile, in a sauté pan, heat the butter and ¼ cup of the pasta water together, whisking to form an emulsified sauce. Add the cooked tortelli and the tarragon to the pan to heat gently and coat with the sauce, about 1 minute. Divide among 6 warmed plate. Finish with the remaining ¼ cup Cacio di Roma cheese and serve immediately.