Place a large pot of water over high heat and bring to a boil.
While waiting for the water to boil in a medium sized mixing bowl, mix the mashed potatoes and goat cheese and form a well. Sprinkle with all the flour, drop the egg and salt in center of well and using a fork, stir into flour and potatoes, just like making normal pasta. Once egg is mixed in, bring dough together, kneading gently until a ball is formed. Knead gently another 4 minutes until ball is dry to the touch. Roll the dough into ¾ of an inch logs and cut into 1 inch pieces. Cover heavily with flour in a single layer (do not layer them or they will become flat and dense). Use immediately or freeze if desired.
Bring a large pot of water to a boil. While waiting for the water to boil in a sauté pan, heat butter over medium heat until the butter begins to brown and take on a nutty aroma. Add shallot, sun dried tomato, thyme, white wine vinegar and half of the Parmesan cheese shaking the pan vigorously to help mix all the ingredients together. Drop the finished dumplings into the water and cook until they float (about 1 minute). Once they float, remove from the boiling water and drop right into the brown butter sauce. Toss a few times to coat the gnocchi and serve on warm plates garnished with the remainder of the Parmesan and chopped parsley.