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1
In a glass measuring cup, mix the whole eggs, egg yolks, olive oil and a pinch of salt.
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2
Put the flour in a food processor.
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3
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.
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4
Transfer the dough to a work surface and knead until smooth and silky, about 2 minutes.
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5
Shape into a ball, wrap in plastic and let stand at room temperature for 15 minutes or refrigerate overnight.
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6
Cut the dough into quarters; work with 1 piece at a time and keep the rest covered.
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7
Flatten the dough slightly and run it through successively narrower settings on a pasta machine until you reach the last setting.
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8
Trim the ends and cut the pasta sheet into 12-inch lengths.
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9
Drape the pasta sheets over a drying rack and let dry for 10 minutes.
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10
Repeat with the remaining dough.
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11
Lightly flour 2 pasta sheets and stack them on top of each other.
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12
Starting at a short end, roll up the sheets very loosely.
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13
Trim the edges and, using a large knife, cut the rolled dough into 1/4-inch-wide strips.
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14
Unroll the strips, toss lightly with flour and transfer to a baking sheet lined with wax paper.
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15
Repeat the process with the remaining sheets of pasta.
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16
Toss the cut pasta occasionally to separate the strands.
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17
Bring a large pot of salted water to a boil.
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18
Add the pasta and cook, stirring occasionally, until barely al dente, 2 to 3 minutes.
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19
Drain the pasta, reserving 1 1/2 cups of the cooking water.
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20
Heat the olive oil in a large, deep skillet until shimmering.
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21
Add the red onion and jalapeno and cook over moderately high heat, stirring, until softened, about 2 minutes.
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22
Add the corn and asparagus stems and tips and cook, stirring occasionally, until crisp-tender, about 3 minutes.
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23
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.
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24
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.
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25
Add 6 tablespoons of the grated Pecorino and toss.
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26
Serve the pasta in shallow bowls, passing extra Pecorino at the table.
