Special equipment: a pastry bag; a fluted ring cutter or fluted pasta roller
For the dough: Mound the flour on a clean, dry work surface and make a big hole in the center of the pile. Crack the eggs into the hole along with the extra yolks, olive oil, water and some salt. Using a fork, beat the eggs. Incorporate the flour, a little at a time, into the egg mixture, being careful not to break the well (or the egg mixture will run everywhere). When enough flour is incorporated that you can handle the dough, use your hands to combine everything really well. (If the mixture is tight and dry, wet your hands a bit.) When the mixture is homogeneous, start kneading. Knead until the dough is smooth and supple, anywhere from 8 to15 minutes. Wrap the dough in plastic wrap and let it rest at room temperature for at least an hour.
For the ravioli: In a food processor, pulse the peas to make a coarse paste. Mix the pea paste, goat cheese, Parmigiano-Reggiano and eggs in a large bowl. Sprinkle with salt. Set aside.
With a rolling pin, roll out the dough into sheets about 1/4-inch thick and 12 inches long, using extra flour as needed to prevent sticking,. (Reserve the excess dough for another use.) Brush the bottom half of the pasta sheets lightly with water.
Fill a disposable pastry bag (or a resealable bag with one corner snipped) with the pea mixture. Pipe 1-inch balls of filling onto the pasta about 1 1/2-inches apart on the bottom half of the pasta sheets. Fold the top half of the pasta sheets over the filling so the top edges meet the bottom edges. Use your index fingers to press around the filling and seal the edges shut. Use a fluted ring cutter or fluted pasta roller to cut out the ravioli. Place the ravioli on a semolina-dusted baking sheet.
Bring a large pot of well-salted water to a boil. Heat the chicken stock and butter in a large saucepan over medium heat; season with salt.
Add the ravioli to the boiling water and cook until they float to the top, 4 to 5 minutes. Transfer the ravioli to the sauce and cook, shaking the pan frequently, until the sauce begins to thicken and cling to the pasta, a few minutes. (If the sauce seems too thick, add a bit more stock.) Place five ravioli on each serving plate, spoon a little extra sauce on top and garnish with pea tendrils, sugar snap peas and mint. Sprinkle with Parmigiano-Reggiano.