1. Make the pasta: Place dough on a lightly floured work surface. Divide dough into 4 equal pieces. Set the rollers of a hand-crank or electric pasta machine at their widest opening. Working with one piece at a time (and keeping the remaining pieces covered with plastic wrap), lightly flour dough and run it through the pasta machine. Fold dough in half and run through machine again.

2. Change rollers of pasta machine to next decreasing setting and roll out dough once without folding. Keep rolling sheet through machine on decreasing settings until you have rolled it through the last (thinnest) setting. Transfer pasta to prepared baking sheet and cover with plastic wrap. Repeat process with remaining pieces of dough.

3. In a medium bowl, mix together ricotta, egg yolks, and Parmigiano cheese; season with salt and pepper. Transfer to a pastry bag fitted with a 1/2-inch round tip; set aside.

4. Working with one sheet of pasta at a time, lay pasta sheet out lengthwise on a lightly floured work surface. Pipe out filling in center of both top and bottom halves of pasta sheet at 1-inch intervals. Brush around filling with beaten egg. Carefully fold bottom edge of sheet over bottom half of filling toward the middle; gently press around filling to seal and remove any air. Repeat process with top half of pasta sheet. Using a pastry cutter, cut raviolini into 1-inch squares around filling and transfer to a parchment paper-lined baking sheet; do not stack, as raviolini will stick. Repeat process with remaining sheets of pasta and filling.

5. Transfer pasta to freezer and freeze until solid; use immediately or transfer to an airtight container and keep frozen up to 2 weeks.

6. Make the melted tomatoes: Meanwhile, heat olive oil in a large skillet over medium-high heat. Add garlic and crushed red pepper flakes; cooking, stirring, until garlic is fragrant, about 2 minutes. Add tomatoes and oregano; season with salt and pepper. Cook until most of the juice from the tomatoes has been released and reduced; about 10 minutes. Add tomato sauce and bring to a boil; reduce heat and stir in basil. Keep warm until ready to serve.

7. Bring a large pot of water to a boil. Generously salt water and return to a boil. Add frozen raviolini and cook until tender, about 1 minute. Meanwhile, melt butter in a medium skillet over medium heat. Drain, reserving 1/4 cup cooking liquid, and add raviolini to skillet with butter. Add reserved cooking liquid and olive oil. Increase heat to medium high and gently toss to combine. Remove from heat and add Parmigiano; toss to combine. Serve immediately topped with melted tomatoes; garnish with more cheese, if desired.