1. Heat a large pot of salted water over high heat. In a large bowl, mash the potatoes until there are no lumps. Fold in the egg yolks and 1/2 teaspoon of the salt and 1/4 cup of the basil. Sift in 1 cup of the flour; knead gently until dough forms. Add remaining flour; knead until dough is soft, not sticky. (Don't overwork dough.) Form into a disk.
2. Transfer dough to a floured work surface and divide into 6 equal pieces. Roll pieces lengthwise until they are 3/4 inch in diameter. Sprinkle with flour and cut crosswise into 1/2-inch pieces. Spread gnocchi on a floured baking sheet and let sit 5 minutes.
3. Heat 1 tablespoon of the olive oil in a large pan over medium heat. Add the onion and saute until soft, 5 minutes. Add the mushrooms and cook 10 minutes.
4. Meanwhile, add gnocchi to boiling water. Cook 3 to 4 minutes, until all the dumplings rise to the surface. Using a slotted spoon (to allow all excess water to drain off), transfer to a bowl and toss with remaining olive oil and salt to taste. Divide among four plates and top with mushroom ragu. Garnish with the remaining basil and Parmesan.