In a large pot of lightly salted water, bring potatoes to a boil; cook until just tender when tested with a fork, about 17 to 20 minutes. Drain potatoes and cool slightly. While still warm, peel potatoes and mash in a medium bowl. Spread mashed potatoes on a baking sheet to cool and allow steam to evaporate.
Meanwhile, bring a large pot of salted water to a boil for cooking gnocchi.
When potatoes are just warm, return them to the bowl. Make a well in centre of potatoes and sprinkle with the flour. Add the eggs. Using a fork, stir together flour and eggs, then gradually incorporate potato mixture until well blended. Season with salt and pepper.
Turn onto lightly floured surface and knead gently but thoroughly to develop a dough that is supple but not sticky, about five to seven minutes, adding flour as needed.
Divide dough into quarters. With your hands, gently roll each quarter into long strands about 2.5 centimeters thick. Using a sharp knife or pastry cutter, cut strand into little rectangles that look like pillows, about 2.5 centimeters. Lay gnocchi on a well-floured baking sheet. Cook right away or freeze immediately.
Boil gnocchi for two to three minutes, or until they rise to the surface and still hold their shape. With a slotted spoon, transfer gnocchi to a bowl.
In a large skillet, melt the butter over medium heat. When the butter is melted and the butter starts to bubble, add garlic and sage. Let the butter foam. Once the butter stops foaming, it will start to brown. Immediately remove the pan from the heat. Add gnocchi and toss with brown butter. Sprinkle with freshly grated pecorino and black pepper. Serve immediately.