Preheat oven to 400 degrees. Dust a baking sheet with flour.
Poke the potato all over with fork. Bake for about 45 minutes, or until fork tender. Remove from oven and let cool for 5 minutes.
Halve the potato and spoon out flesh in to a large bowl, discarding the skin. Add pumpkin puree and mash together until mostly smooth. Add the eggs, flour, pecorino, salt, and pepper. Still until just combined. If the dough seems wet, add a little more flour. The dough should be sticky. Form in to a ball.
Scrape dough onto generous floured working surface. Cut the dough in to 4 equal pieces. One at a time, roll dough in to a 1 inch rope. Using a sharp knife, cut in to 1 inch pieces. Place gnocchi on to prepared baking sheet. Repeat with remaining sections of dough.
Bring large pot of water to boil over high heat. Add gnocchi and cook until they float to the top, 3-4 minutes. Remove with slotted spoon.
In a large skillet, heat olive oil over medium high. Add gnocchi in a single layer, working in batches if necessary, and cook, undisturbed, until golden and crisp, 3-4 minutes. Turn the gnocchi and cook 3 minutes more. Remove for skillet and set aside.
In the same skillet, immediately add the butter and oregano and cook over medium heat until the butter is browned and the oregano is crisp. Remove from the skillet from heat and gently stir gnocchi in to the butter sauce.
Serve immediately with the manchego.