Boil the whole potatoes until they are soft (about 45 minutes). While still warm, peel and pass through vegetable mill onto clean pasta board.
Bring 6 quarts of water to boil in a large spaghetti pot. Set up an ice bath with 6 cups ice and 6 cups water near the spaghetti pot.
Make a well in center of potatoes and sprinkle all over with flour, using all the flour. Place egg and salt in center of well and using a fork, stir into flour and potatoes, just like making normal pasta. Once egg is mixed in, bring dough together, kneading gently until a ball is formed. Knead gently another 4 minutes until ball is dry to the touch.
Roll a baseball-sized ball of dough into 3/4-inch diameter dowels and cut dowels into 1-inch long pieces. Flick pieces off a fork or concave side of cheese grater until dowel is finished. Drop these pieces into boiling water and cook until they float (about 1 minute). Meanwhile, continue with remaining dough, forming dowels, cutting into 1-inch pieces and flicking off fork. As gnocchi float to top of boiling water, remove them to the ice bath. Continue until all have been cooled off. Allow the gnocchi to sit several minutes in the ice bath before draining them from the ice and water. Toss the cooled gnocchi with 1/2 cup canola oil and store covered in refrigerator up to 48 hours until ready to serve.
In a saute pan, heat 1 tablespoon butter over medium-high heat until it foams. Add the chopped shallots and cook until the shallots are soft and translucent. Add the mushroom and let sit, without stirring until they give out their juices. Toss the mushrooms and season to taste with salt and pepper. Add the chopped thyme, and continue cooking until the mushrooms are soft and golden brown, about 7 minutes more.
Add the remaining butter and 1/4 cup of the cooking water (if serving with pasta or gnocchi). Whisk together as the butter melts to form an emulsified sauce. Add the cooked gnocchi to the pan once the sauce is emulsified and hot. Toss quickly to combine and heat though, about 1 minute, then serve immediately.
While your pasta cooks, melt the butter in a 12 to 14-inch saucepan until it foams and subsides. Add 2 to 4 tablespoons of the pasta cooking water and whisk to emulsify the sauce. Add the sage leaves and toss for 1 minute to imbue the butter with the flavor of the sage. Drain the pasta and add the pan. Toss over high heat to coat the pasta with the sauce and serve immediately.