Bring a large pot of water to a boil and salt it. Add potatoes and cook 7 minutes. Add the pasta and cook 2 minutes, then stir in cabbage and cook 5 minutes longer.
While the potatoes and pasta cook, melt butter in a large, deep skillet over medium heat and add garlic and 6 whole sage leaves. Cook 3 to 4 minutes; remove the garlic and whole leaves (reserve the leaves for ganish), then add sliced leaves and the black pepper. Just before draining the pasta and cabbage, add 2 ladles of starchy cooking water to the skillet.
Drain the potatoes, pasta and cabbage and add to butter and sage. Toss to combine with sage butter, adding in lots of cheese as you work. Toss 1 to 2 minutes to form a cheesy butter coating on the pasta and cabbage then adjust salt and serve. Use reserved whole leaves to garnish the pasta.