Heat 2 tablespoons olive oil in a medium saucepan over medium heat. Add the garlic and cook, stirring, until softened, 1 to 2 minutes. Stir in the tomatoes and mash into a chunky sauce using a wooden spoon. Bring to a boil, then reduce the heat to a gentle simmer and cook, stirring occasionally, until the sauce thickens, about 12 minutes. Stir in the basil and season with salt and pepper. Remove from the heat and transfer to a blender; puree.
Dust a rimmed baking sheet with flour; set aside. Mix the spinach, ricotta, parmesan, egg yolks, nutmeg and 1 teaspoon each salt and pepper in a large bowl. Add the flour and stir until just combined. Drop tablespoonfuls of the dough onto the flour-dusted pan, shaking the pan from side to side to coat the dumplings in the flour and smooth out their edges. Refrigerate 10 minutes.
Meanwhile, bring a large pot of water to a boil. Reduce the heat to maintain a steady simmer. Add the chilled dumplings and cook until they float to the top and are tender, 4 to 5 minutes. Remove with a slotted spoon and transfer to a paper towel¿lined plate to drain. Serve on top of the tomato sauce and drizzle with the remaining 2 tablespoons olive oil. Sprinkle with more parmesan.