Preheat the oven to 375 degrees F. Scrub the potatoes well and pierce in several places with the tip of a knife. Bake on a baking sheet until cooked through, about 1 hour. Set aside to cool.
Once they're cool enough to handle, peel the potatoes and pass the flesh through a ricer or the finest holes of a food mill. Place 1 1/2 cups of the flour on a clean work surface and put the potatoes in the middle. Season with the salt. Knead the potatoes and the flour together until it's a smooth ball of dough that's slightly sticky. If the dough is too sticky, add more of the remaining 1/2 cup flour gradually to achieve the smooth consistency.
Divide the dough into 4 equal pieces and cover 3 of them with a dampened kitchen towel. On a lightly floured work surface, roll the uncovered piece of dough into a 1-inch-thick rope. Cut into 3/4-inch pieces. Turn a fork over and hold at a 45-degree angle, with the tines touching the work surface. Working with one piece of dough at a time, roll it down the fork tines, pressing with your thumb to make ridges on one side. Place on a large baking sheet lined with a kitchen towel. Repeat with the remaining pieces of dough.
Bring a large pot of salted water to a boil. Add half the gnocchi and cook until they bob to the surface of the water. Transfer to a shallow bowl and cook the remaining gnocchi.
Serve the gnocchi warm with the Marinara Sauce. Top with shaved Parmesan and garnish with chopped basil.
Heat the olive oil in a large skillet over medium-high heat until starting to shimmer. Add the garlic and cook, stirring, until fragrant, 1 to 2 minutes. Add the chopped tomatoes, Italian seasoning, and salt and pepper to taste. Bring to a boil and cook for 5 minutes. Stir in the butter until melted. Remove from the heat and add the chopped basil.