Cook the butter in a heavy medium skillet over medium heat until it begins to brown, about 2 minutes. Remove from the heat. Add the thyme leaves. Set aside.

Pierce the potato all over with a fork. Microwave the potato until tender, turning once, about 12 minutes. Cut the potato in half and scoop the flesh into a large bowl; discard the skin. Using a fork, mash the potato well. Mash in the salt and pepper. Mix in 3 tablespoons of the egg; discard the remaining egg. Sift the flour over the potato mixture and knead just until blended.

Divide the dough into 4 equal pieces. Roll each piece between your palms and the work surface into a 1/2-inch-diameter rope (about 20 inches long). Cut the dough into 1-inch pieces. Roll each piece of dough over a wooden paddle with ridges or over the tines of a fork to form grooves in the dough.

Cook the gnocchi in a large pot of boiling salted water until the gnocchi rise to the surface, about 1 minute. Continue cooking until the gnocchi are tender, about 4 minutes longer. Using a slotted spoon, transfer the gnocchi to the hot thyme-butter in the skillet. Toss to coat.

Spoon the gnocchi and butter sauce into shallow bowls. Top with the Pecorino and serve.