Bring a large pot of water to a boil; salt it. Add the potatoes and cook for 7 minutes. Add the pasta and cook for 2 minutes. Stir in the cabbage and cook for 5 minutes.
While the potatoes and pasta are working, melt the butter in a large, deep skillet over medium heat. Add the garlic and whole sage leaves. Cook until the sage is crisp, 3 to 4 minutes. Remove the garlic and sage leaves to a small plate.
Add the sliced sage and ground pepper to the skillet; just before draining the potatoes, pasta and cabbage, add 2 ladles of the starchy cooking water.
Drain the potatoes, pasta and cabbage and add to the skillet. Stir it all together, adding the Pecorino Romano as you work to get a cheesy, buttery coating. Adjust the salt and garnish the pasta with the reserved whole sage leaves and garlic.