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