Meanwhile, bring a large pot of salted water to a boil. Add the potato and cook until tender, about 5 minutes. Using a slotted spoon, transfer the potato to a bowl. Add the pasta to the pot and cook, stirring, until al dente, about 2 minutes. Using a slotted spoon, transfer the pasta to a colander. Add the cabbage to the pot and cook until just wilted, about 3 minutes. Drain the cabbage and shake out the excess water. Pour off the water and return the cabbage to the pot, along with the pasta and potato. Add the browned butter, season with salt and pepper and toss well. Stir in 1 cup of the Fontina and half of the Parmigiano and transfer the mixture to a 9-by-13-inch baking dish.