2 1/4 cups All-Purpose Flour (plus more for dusting)
3 Extra Large Eggs
6 Extra Large Egg Yolks
HOW TO MAKE THIS RECIPE
Basic Pasta Dough: Put the flour, eggs, and egg yolks in the bowl of a standing mixer fitted with the paddle attachment and mix on low speed until the dough comes together. Turn off the mixer, remove the paddle attachment, and replace it with the dough hook. Scrape down the sides of the bowl and beat the dough with the dough hook on medium speed until it forms a ball, about 5 minutes. Dust a flat work surface with flour. Turn the dough out onto the dusted surface and gently knead it for 20 to 25 minutes, until the ball begins to feel elastic and the surface of the dough feels smooth and silky. Wrap the dough in plastic wrap and refrigerate to rest for at least 45 minutes and up to overnight before sheeting it (any longer and the dough will discolor).
To make the ravioli, cut eight 4-inch-square pieces of parchment and dust two baking sheets with semolina. Roll the dough out to the third thinnest setting on a pasta sheeter, stacking the sheets of dough on one of the parchment-lined baking sheets. When you have rolled out all of the dough, dust a flat work surface with flour and lay one sheet of the dough on the floured surface. Place three or four ricotta disks on the dough, leaving 4 inches between each disk and 1 1/2 to 2 inches of dough on all the sides of the outer disks. Use the back of a spoon or your fingers to make a crater deep enough to hold an egg yolk in the center of each disk. Separate one egg, reserving the white and carefully sliding the yolk into one crater. Repeat, filling the remaining 2 or 3 disks in the same way (you don't need to reserve any more whites). Lay a second sheet of dough directly behind the first and use a rolling pastry cutter, starting at the edge of the dough nearest your body and moving in a straight line directly away from you, making one long cut through both sheets of dough between two disks of ricotta. Continue making cuts down the sheet of dough between the disks. Using a pastry brush, brush egg white around each cheese disk. Lift one segment of dough from the back sheet and lay it on top of the corresponding front segment of dough. Without lifting the raviolo, cup both hands around the cheese so that the wedges of your pinky fingers press down around the cheese, sealing the raviolo closed. Repeat, covering and sealing the remaining ravioli. Use a 3x3" fluted cookie cutter or fluted pastry wheel to cut each raviolo, discarding the scraps of dough around it. Place the ravioli on the prepared baking sheet and repeat, assembling the remaining ravioli in the same way. Cover the baking sheet with plastic wrap and refrigerate until you are ready to boil them, or for up to 8 hours.
Fill two wide pots with 6 quarts of water each. Add 6 tablespoons of salt to each pot and bring to a boil over high heat. Have a slotted spatula or slotted spoon and a clean dishtowel handy for lifting the ravioli out of the water.
Remove the ravioli from the refrigerator and, one at a time, lift the parchment holding each raviolo off the baking sheet and gently drop the raviolo with the paper into the water, adding 4 ravioli to each pot. (The parchment will quickly separate and will be easy to remove from the water.) When you have dropped all the ravioli, remove and discard the parchment paper, partially cover the pots to return the water to a boil quickly, and keep it boiling. Cook the ravioli for 4 minutes.
With a clean dishtowel in one hand and a slotted spoon or spatula in the other, lift one raviolo out of the water and onto the dishtowel to blot it dry, and then carefully place it in the center of a dinner plate. Repeat with the remaining ravioli. Pour the brown butter over the ravioli, dividing it evenly. Place one of the sage leaves on top of each ravioli discarding the remaining leaves. Use a microplane of another fine grater to grade a light dusting of Parmigiano-Reggiano over each raviolo, and serve.
Brown Butter Sauce: While the water is coming to a boil, make the brown butter sauce. Place the butter and sage leaves in a medium saucepan and cook over medium-high heat for 3 to 5 minutes without stirring, swirling the pan occasionally to brown the butter evenly and prevent it from burning, until the bubbles subside and the butter is dark brown with a nutty, toasty smell. Reduce the heat to low to keep the butter warm while you cook the pasta.