Make the gnocchi. Mix the mashed sweet potato, pureed burrata or ricotta, eggs, salt and parmesan together in a large bowl. Add 2 cups of the flour and mix well with your hands. The dough should be very sticky. Add another half cup of flour and mix in. You want the dough to still be pretty sticky, but sturdy enough to shape into a ball. If it's not, keep adding a little flour at a time until you can get a soft dough that will be rollable.
Spread some flour on a large work surface. Cut the dough log into four equal pieces. Take one piece and cut it in half. Roll the piece of dough into a snake about 1/2 inch thick, then cut it into pieces about the width of a fork. Dust the gnocchi with a little flour, then use one finger to push the gnocchi up onto the tines of a fork. Let the gnocchi drop back to the work surface. Doing this helps create ridges for the sauce to stick to, but you can skip it if you would like. Repeat this process with the other piece of dough and place the gnocchi on a large plate, cover and set aside.
Bring a large pot of salted water to a boil.
To make the mushroom and brown butter sauce. In a medium skillet over high heat, add 2 tablespoons butter. When the butter is melted, sprinkle in the mushrooms in a single layer. Don't stir them! Let them sizzle until they have caramelized on the bottom, about 2 minutes. When the bottoms are caramelized, toss them once and season with salt and pepper, to taste. Continue to cook without stirring for about 5 minutes. Season mushrooms with salt and pepper. Add the remaining butter and cook until it begins to brown. Once the butter is browned reduce the heat and add the garlic, thyme and rosemary and cook for about 10 seconds. Add the balsamic vinegar, and simmer until the mushrooms are glazed with the sauce. Then stir and remove the pan from the heat and slide the mushrooms and sauce off to the side.
Now grab your gnocchi and add it to the salted boiling water. Boil the gnocchi until they float, then remove them with a slotted spoon and add them right into the skillet with the mushrooms/sauce. Return the skillet (with the mushrooms and gnocchi in it) back to medium heat. Let the gnocchi get crisp on one side for 2 minutes and then two minutes on the other, then stir gnocchi into the mushrooms. Remove from the heat and serve immediately with crumbled goat cheese on top. EAT!