Arrange a rack in the center of the oven and preheat to 350 degrees F. Wash the potatoes, pat dry and rub with olive oil. Put the oiled potatoes on a baking sheet, sprinkle with salt and bake until the potatoes are easily pierced with a small knife, 45 to 60 minutes. Set aside to cool. This may be done up to a day in advance; keep the potatoes in the refrigerator until ready to use.
Peel the potatoes with a paring knife – the skin should easily pull away – and cut into halves or quarters. Pass the potatoes through a ricer or food mill into a large bowl or coarsely grate with a cheese grater.
Add the flour, Parmesan and 1 teaspoon salt to the riced potatoes. Gently stir to combine the ingredients but retain the crumbly texture. Create a well in the center of the bowl and crack in the eggs. Stir the eggs with a wooden spoon, then gradually incorporate the potato mixture (do not overmix).
When the mixture forms a ball, turn it out onto a lightly floured surface and gently knead for 1 minute, then divide into 2 pieces (or cover tightly and refrigerate until ready to use, up to 24 hours).
Take a piece of the dough and roll it into a "snake" about 1 inch in diameter. Cut the snake into pieces about 1 1/2 inches in length, sprinkle lightly with flour, and then roll each one over a gnocchi paddle or the backside of a fork to lightly mark with ridges. Continue until all the gnocchi are formed (keep them on a floured baking sheet as you go). Repeat the rolling and cutting process with the remaining gnocchi dough.
Bring a large pot of salted water to a rapid boil.
Add the oil and garlic to a large skillet over medium heat, placed next to the pot of water. Add the garlic and cook until aromatic, about 1 minute. Add the shrimp and cook until lightly pink on both sides but not cooked through, 1 to 2 minutes (the shrimp will finish cooking in the sauce). Remove the shrimp to a bowl.
Add the tomatoes to the pan, bring to a boil and cook for 5 minutes. Add the shrimp back to the pan and stir to combine. Transfer the sauce to a larger pan or pot so there is enough space for the cooked gnocchi.
Drop the gnocchi into the boiling water and gently stir to prevent sticking. Cook until all the pieces rise to the surface, 2 to 3 minutes.
Transfer the cooked gnocchi to the pan of tomato sauce along with about 1/4 cup of the pasta water. Cook on high for 1 minute, stirring gently. Remove from the heat and stir in the lemon zest and basil.