On a clean counter, mound the flour and make a wide well. Add eggs, olive oil, and salt to the center.
Whisk the eggs with a fork, pulling in flour from the inner rim until a thick paste forms. Switch to your hands and knead.
Knead 8–10 minutes until the dough feels smooth, elastic, and slightly tacky. If it crumbles, add water 1/2 teaspoon at a time. If it sticks, dust lightly with flour.
Wrap and rest 30 minutes at room temperature. This relaxes the gluten so you can roll thin.
In a bowl, combine ricotta, mozzarella, Parmesan, basil, parsley, garlic, egg yolk, pepper, salt, and red pepper flakes if using.
Stir until smooth. Taste and adjust salt. The mixture should hold its shape, not run.
Transfer to a piping bag or a zip-top bag and chill while you roll the dough.
Divide dough into 4 pieces. Keep pieces covered so they don’t dry out.
With a rolling pin: Dust the counter lightly and roll each piece into a long, thin sheet—about 1/16 inch (card-stock thin). Or use a pasta machine down to a thin setting.
Lay sheets on a lightly floured surface. If they feel sticky, dust with flour; if they look dry, brush lightly with water.
Pair sheets of similar size. On one sheet, pipe or spoon 1-teaspoon mounds of filling spaced about 1.5 inches apart.
Brush around each mound with a little water. Lay the second sheet over the top, pressing gently around each mound to push out air.
Cut between mounds with a knife or pastry wheel to form squares. Press edges firmly; crimp with a fork for a tight seal.
Set ravioli on a floured sheet pan. Dust lightly and cover with a towel while you finish the batch.
Bring a large pot of generously salted water to a strong simmer. Avoid a hard boil, which can jostle and split ravioli.
Melt butter in a skillet over medium heat. Add sage and let the butter foam and turn golden with brown specks, 2–3 minutes. Remove from heat.
Slide in ravioli in batches. Cook 2–4 minutes until they float and feel tender. Lift gently with a spider or slotted spoon.
Toss ravioli in the browned butter. Finish with more Parmesan and a drizzle of olive oil if you like.
