How to Make It
In a glass measuring cup, mix the whole eggs, egg yolks, olive oil and a pinch of salt. Put the flour in a food processor. Drizzle the egg mixture over the flour and process in long pulses until the dough is evenly moistened and forms small clumps, about 2 minutes. Transfer the dough to a work surface and knead until smooth and silky, about 2 minutes. Shape into a ball, wrap in plastic and let stand at room temperature for 15 minutes or refrigerate overnight.
Cut the dough into quarters; work with 1 piece at a time and keep the rest covered. Flatten the dough slightly and run it through successively narrower settings on a pasta machine until you reach the last setting. Trim the ends and cut the pasta sheet into 12-inch lengths. Drape the pasta sheets over a drying rack and let dry for 10 minutes. Repeat with the remaining dough.
Lightly flour 2 pasta sheets and stack them on top of each other. Starting at a short end, roll up the sheets very loosely. Trim the edges and, using a large knife, cut the rolled dough into 1/4-inch-wide strips. Unroll the strips, toss lightly with flour and transfer to a baking sheet lined with wax paper. Repeat the process with the remaining sheets of pasta. Toss the cut pasta occasionally to separate the strands.
Bring a large pot of salted water to a boil. Add the pasta and cook, stirring occasionally, until barely al dente, 2 to 3 minutes. Drain the pasta, reserving 1 1/2 cups of the cooking water.
Heat the olive oil in a large, deep skillet until shimmering. Add the red onion and jalapeño and cook over moderately high heat, stirring, until softened, about 2 minutes. Add the corn and asparagus stems and tips and cook, stirring occasionally, until crisp-tender, about 3 minutes.
Add 1 cup of the reserved pasta cooking water and a generous pinch of salt and pepper to the vegetables and bring to a boil over moderately high heat. Add the pasta and cook, stirring, until the liquid is nearly absorbed and the pasta is tender, about 2 minutes; if necessary, add a little more of the pasta cooking water. Add 6 tablespoons of the grated Pecorino and toss. Serve the pasta in shallow bowls, passing extra Pecorino at the table.