To make gnocchi: Mix sweet potato, ricotta, eggs, salt and ¼ cup Parmesan together in a large bowl. Add 2½ cups of the flour, and mix well with your hands. Dough will be sticky but should be sturdy enough to shape into a ball. Keep adding a little flour at a time until you get a soft dough.
Spread some flour on a large work surface. Roll dough into a log, and cut into four equal pieces. Take one piece, and cut in half. Roll piece into a log shape about ½-inch thick; then cut into pieces about the width of a fork. Dust gnocchi with a little flour; then use one finger to push gnocchi up onto the tines of a fork. Let gnocchi drop back to the work surface. Repeat with all pieces.
Bring a large pot of salted water to a boil.
While waiting for water to boil, in a medium skillet over high heat, add 2 tablespoons butter. When butter is melted, sprinkle in mushrooms in a single layer. Let caramelize on the bottom, about 2 minutes. When caramelized, toss once and season with salt and pepper, to taste. Continue to cook without stirring, 5 minutes.
Add remaining butter, and cook until it begins to brown. Once butter is browned, reduce heat and add garlic, thyme, rosemary and red pepper flakes. Sauté about 1 minute; then add balsamic vinegar. Simmer until mushrooms are glazed with sauce. Remove from heat.
Add gnocchi to boiling water. Boil until they float; then remove with a slotted spoon, and add into skillet with sauce. When all gnocchi are placed in saucepan, return skillet to medium heat. Let gnocchi crisp on one side; then stir around to mix in mushroom sauce. Remove from heat, and serve immediately with remaining Parmesan and freshly ground black pepper.