1. Preheat the oven to 400°. Pierce the potatoes all over with a fork. Bake in a microwave oven at high power for 10 minutes, then flip the potatoes and microwave for 5 minutes longer. Transfer the potatoes to the oven and bake for 15 minutes. Alternatively, bake the potatoes in the oven for about 1 hour, until tender.
2. In a skillet, cook the carrots in the olive oil over moderate heat for 2 minutes. Add the water and a pinch of salt. Cover and cook until tender, 15 minutes. Puree in a food processor.
3. Halve the potatoes. Scoop the flesh into a ricer and rice the potatoes. Transfer 2 slightly packed cups of riced potatoes to a bowl. Stir in the egg yolks, 1/2 cup of the carrot puree and 1 teaspoon of salt. Add the 1/2 cup and 2 tablespoons of flour; stir until a stiff dough forms. Knead the dough gently until smooth but slightly sticky.
4. Line a baking sheet with wax paper and dust with flour. On a floured surface, cut the dough into 4 pieces, rolling each into a 3/4-inch-thick rope. Cut the ropes into 3/4-inch pieces. Roll each piece against the tines of a fork to make ridges; transfer to the baking sheet.
5. In a large, deep skillet of simmering salted water, cook the gnocchi until they rise to the surface, then simmer for 2 minutes longer. In a large nonstick skillet, melt the butter. Using a slotted spoon, add the gnocchi to the butter. Season with salt and pepper and cook over high heat for 1 minute. Sprinkle with the cheese and serve.