Make the dough: In a large bowl or on a clean countertop, add the flour and a pinch of salt. Create a well in the center and add the whole eggs and oil; starting in the center, mix with a fork, beating the eggs and gradually incorporating more flour until the dough is a sandy consistency. Add the egg yolks and continue stirring. Using your hands, knead the dough until it forms a shaggy ball, then turn it out onto a counter and continue kneading until it forms a smooth and springy ball, a full 5–7 minutes. Wrap tightly in plastic and let rest for 20 minutes. (After resting, dough should be smooth and not tacky.)
Meanwhile, prepare the filling: In a large skillet, heat the olive oil over medium-high heat. Add the kale and collard leaves and season with a pinch each of salt and pepper; cook, stirring frequently, until tender with a little brown on the edges, about 4 minutes. Stir in the garlic and let cook until softened but not browned, 1 minute. Transfer the contents of the skillet to a large bowl and let cool completely.
Prepare a large pot of gently boiling water. Meanwhile, pat the cooled greens dry to take away any excess liquid. Finely chop them (you should have 1 cup). Dry out the bowl and return the greens to the bowl. Add the rest of the filling ingredients and mix until combined. Refrigerate until ready to use.
Make the sauce: In a large, clean skillet, add the olive oil. Add the garlic and cook, turning occasionally, until very lightly browned, about 2 minutes. Add the rosemary, ½ teaspoon fresh oregano, and the chili flakes and heat until just fragrant (be careful not to burn), about 1 minute. Pour in the tomatoes all at once and bring to a simmer; cook until the consistency is thickened slightly, 9-10 minutes. Turn off the heat and set aside until ready to use.
Roll out the pasta: Dust a large baking sheet lightly with flour. Using a pasta machine or rolling pin, roll out pieces of the dough into thin sheets, keeping the remainder of the dough wrapped in plastic (use setting #7 on a pasta machine to roll out pieces of the dough, or roll pieces of the dough using a rolling pin until they are very thin without being at all translucent). If using a pasta machine, cut the long dough strips in half crosswise. Working on one of the resulting strips, dollop 1 tablespoon of the filling about 1 inch away from the short end of the dough into its center. Continue dolloping compact balls of filling down the dough about 3 inches apart from each other. Place the other rolled sheet of dough on top and tuck the dough around the edges of the filling, tightly packing the filling into a neat circle. Using a 2¾-inch or 3-inch round ravioli, biscuit, or cookie cutter, cut out the ravioli by pressing the cutter down firmly and twisting slightly. Pull away the surrounding dough and save to re-roll. Pick up each ravioli and press the edges to seal, stretching them ever so slightly. Set the ravioli on the prepared baking sheet and continue with the remaining dough and filling.
Season the boiling water generously with salt and bring back to a gentle boil. Add the ravioli and cook until they are al dente and float to the top on their own, about 3 minutes. Meanwhile, reheat the sauce if needed (it should be at a very bare simmer). Using a slotted spoon or spider tool, transfer the ravioli to the sauce, gently submerging them. Add about 2 tablespoons of the salty pasta water, or a little more as needed to thin the sauce, and add the butter. Incorporate the butter, season to taste, and fold in half of the fresh oregano. Serve the ravioli and top with parmigiano and the remaining oregano.