Peel potatoes and cut into 3/4-in. pieces. Put in a large pot, add 2 qts. water, cover, and bring to a boil. Add salt and linguine. Stir, cover, and return to a vigorous boil. Add green beans, cover, and bring back to a boil. Uncover and cook until linguine is tender to the bite, about 5 minutes.
Meanwhile, chop sage leaves. In a large serving bowl, mash cheese, butter, sage, and pepper together. Set aside.
Reserve 1/2 cup pasta water and set aside. Drain pasta and vegetables, shaking off as much water as possible. Pour pasta and vegetables on top of cheese-butter mixture. Toss to combine until cheese melts and coats pasta. If resulting sauce is too thick, add reserved pasta water, 1 tbsp. at a time. Serve hot, garnished with additional sage or pepper if you like.
Spinach: Add 6 oz. of spinach, cut into thin ribbons or chopped, to the pasta for the last minute of cooking.
Radicchio: In a large frying pan over high heat, cook 1 head thinly sliced radicchio in 2 tsp. olive oil until wilted, about 3 minutes. Add to cooked pasta before tossing with cheese.
Herbs: Use thyme, oregano, or basil instead of sage.