How to Make It
In a small bowl, whisk the eggs with the egg yolks and 1/4 cup of water. Mound the flour on a work surface, make a well in the center and add the eggs. Using a fork, gradually draw in the flour until a dough forms. Gather and knead the dough until smooth. Form into a ball, cover in plastic wrap and refrigerate for 1 hour.
In a food processor, pulse the feta until smooth. Scrape into a medium bowl and add the next 5 ingredients. Mix until well blended and season with salt. Refrigerate until ready to use.
Grate the tomato halves on the large holes of a box grater set in a bowl until only the skins remain; discard the skins. In a large saucepan, heat the olive oil. Add the garlic and cook over moderate heat, stirring, until fragrant, about 2 minutes. Add the grated tomatoes and crushed red pepper and season with salt. Cook over moderate heat, stirring occasionally, until the sauce has thickened and reduced to 3 cups, about 20 minutes.
Divide the pasta dough into 4 pieces. Work with 1 piece at a time, keeping the rest covered. Press the dough to flatten it slightly. Run the dough twice through a pasta machine at the widest setting. Run the dough through 4 successively narrower settings, twice per setting, until the dough is about 1/16 inch thick. Dust a work surface with semolina and lay the pasta sheet on top; dust the pasta generously with semolina and cover with a damp kitchen towel. Repeat with the remaining 3 pieces of pasta dough, keeping the rolled-out sheets covered.
Lay one of the pasta sheets on a work surface and brush off the semolina. Spoon four 1/4-cup mounds of the cheese filling onto the pasta sheet about 4 inches apart. Spread the mounds of filling into 3-inch rounds and make a small well in the center of the filling. Slip 1 egg yolk into each well. Brush around the filling with the beaten egg whites. Drape another sheet of pasta on top and press around the fillings to push any air pockets out of the ravioli. Using a 4-inch fluted cookie cutter, stamp out the ravioli. Press the edges to make sure they are sealed. Transfer the ravioli to a semolina-dusted baking sheet. Repeat to form another 4 ravioli; reserve any remaining filling for another use.
Bring a large pot of water to a boil and season generously with salt. Reheat the tomato sauce in the saucepan. Using a slotted spoon, add 4 of the ravioli to the boiling water and cook until the pasta is al dente, about 3 minutes. Using a slotted spoon, transfer the ravioli to the tomato sauce. Add half of the torn basil and cook, stirring gently, until nicely coated, about 1 minute. Transfer the ravioli to plates and spoon some of the sauce on top. Repeat with the remaining 4 ravioli and torn basil. Garnish with Pecorino Romano and basil leaves and serve.