Boil the potatoes in a large pot of water until soft, about 30 to 40 minutes. Peel and mash the potatoes (gently, by hand).

Sift the flour out onto a counter or workspace and arrange it in a mound. Add the potatoes to the top of the mound, then add the egg white, grated cheese, and a dash of salt. Knead the dough with your hands until you get a soft and pliable dough.

Split the dough in half. Take each half and roll it out with your hands into a long "snake" about 2 centimeters in diameter. Cut the rolls of dough into 1-inch pieces, then roll them with a fork (or simply roll them on your thumb) to get the classic gnocchi shape.

Bring a pot of salted water to a boil. Add the gnocchi and boil until they float to the top - two to three minutes. Drain the gnocchi and prepare to serve immediately with the tomato sauce.

In a medium saucepan, heat the extra virgin olive oil. Add the minced onions and saute briefly, just until softened. Stir in the tomato puree, bring to a boil, and simmer for 15-20 minutes, adding water as necessary to keep the sauce to the consistency of a thick gravy.

Add the cooked, drained gnocchi to the tomato sauce, along with a few basil leaves, and stir, allowing the gnocchi to simmer for two to three minutes.

Divide the gnocchi evenly into four small ovenproof ramekins. Distribute the cheese evenly over the gnocchi. Bake at 400 degrees for five minutes or until the cheese is melted and bubbly. Top with remaining basil and serve.