To make the gnocchi: place the potatoes in a large pot. Cover with water. Add salt and bring the water to a boil over high heat. Reduce the heat to medium and simmer the potatoes until they are tender, about 35 minutes. Drain the potatoes. When they are just cool enough to handle, peel them then pass them through a food mill.
In a large bowl, mix the flour with the egg yolk, cheese and warm potatoes. Work the mixture until it becomes a smooth dough, then incorporate the herbs. Form the dough into 2 balls.
On a lightly floured work surface, roll each ball of dough into a 'rope' about 1/2-inch thick. Cut the 'ropes' into 3/4-inch pieces. Arrange the gnocchi on a lightly floured baking sheet. (You can freeze the gnocchi at this point and use straight from freezer.)
Shortly before serving, bring a large pot of salted water to boil over high heat. Reduce the heat so the water maintains an active simmer. Working in small batches, drop gnocchi into the water. Cook the gnocchi until they float 3 to 4 minutes, then remove with a slotted spoon and drain fully in a colander.
To make the sauce: Melt the butter in a large skillet over medium heat. Add the shallots and cook, stirring occasionally, until they soften. Add the vodka, cream, and tomato sauce, and simmer until the flavors blend, 2 to 3 minutes.
Immediately before serving, add the gnocchi to the sauce and heat, stirring gently to coat. Spoon the gnocchi into warm bowls and serve.
Puree whole tomatoes without their liquid in a food mill. Heat the oil in a large deep skillet over medium heat. Add the pureed tomatoes. Add the basil, season with salt and pepper and simmer, stirring frequently until the sauce is concentrated, about 25 minutes. Remove the basil and refrigerate or freeze until ready to use.
This recipe was provided by a chef, restaurant or culinary professional. It has not been tested for home use.