For the Simple Sauce
⅓ cup extra-virgin olive oil
¼ cup garlic, sliced
2 cups scallions , thinly sliced, whites and greens
½ teaspoon salt, plus more if needed
freshly ground black pepper
boiling water from the pasta pot
For the Frittata
6 large eggs
¼ cup milk
½ teaspoon salt
freshly ground black pepper
¼ cup extra-virgin olive oil
For the Pasta and Finishing
1 pound spaghetti
2 tablespoons extra-virgin olive oil
1 cup Parmigiano-Reggiano, freshly grated
1½ tablespoons kosher salt, for the pasta cooking water
Start heating 6 quarts of water with 1-1/2 tablespoon kosher salt in the pasta cooking pot.
Have all the ingredients for the simple sauce measured and at hand. Have both skillets on the stove, ready to start cooking.
For the frittata, snap off the tough bottom stubs of the asparagus and peel the thick skin from the lower 4-inches of each spear. Slice the spears, including the tips, crosswise or slightly on the diagonal, into 1/8-inch thick rounds or ovals. You should have 2 cups of small asparagus pieces. Crack the eggs into a bowl and whisk them well with the milk, 1/2 teaspoon of salt and several grinds of black pepper.
Pour 1/3 cup of the olive oil into the big pasta skillet, scatter the garlic slices in the oil and set on medium heat. Cook for minutes or so, shaking the pan now and then, until the garlic is sizzling and starting to color.
Toss the scallions into the skillet, stir with the garlic and oil and cook for 2-1/2 to 3 minutes, over medium heat, until the scallions are very soft and wilted and the garlic is golden. While the scallions are cooking, stir in 1/2 teaspoon of salt and a generous amount of freshly ground black pepper.
Take 2 cups of water from the pasta pot, pour it into the skillet,. Stir up the scallions and garlic and adjust the heat so the liquid is simmering gently.
Pour the 1/4 cup of olive oil into the empty nonstick skillet, scatter in the sliced asparagus and turn on medium heat. Let the pieces heat and start to sizzle, about 2 minutes; stir and toss now and then.
As soon as you've started the asparagus, push the spaghetti into the boiling water. Stir as it softens and bring the water back to the boil as rapidly as possible. Start timing the pasta to undercook as required, about 8 minutes.
Now take 1/4 cup or so of water from the pasta pot and pour it in with the sizzling asparagus. Season with grinds of black pepper and 1/4 teaspoon of salt. Stir toss and cook the slices until they're cooked through but still bright green, about 4 minutes. Moisten with more pasta water if the slices seem dry.
While the spaghetti and the asparagus cook, make sure there's a good layer of simmering liquid in the scallion sauce skillet; add more pasta water if needed.
When the asparagus slices are tender, turn down the heat a bit and pour in the eggs. Let them set for about 1/2 minute then stir thickening eggs and the asparagus with a wooden spoon. Continue turning the curds over for a minute so they are evenly heated but still quite wet and soft. Remove from the heat.
Check the spaghetti: when it is almost but not quite al dente, lift the pasta from the pot and drop it into the simmering simple sauce.
Raise the heat if it was very low, add more hot pasta water if the pasta seems dry. Toss and cook the spaghetti and scallion sauce together for 1/2 minute, mixing well.
Scrape the frittata on top of the pasta and toss it in, breaking the frittata into small pieces. If any eggs are stuck in the pan, loosen them with a slosh of pasta water and toss that into the pasta skillet too. Toss for a minute or more, still over heat, until the pasta is perfectly cooked and the frittata is well distributed.
Remove the skillet from the heat, drizzle a tablespoon or 2 of olive oil over and toss. Sprinkle on the grated cheese and toss. Grind black pepper on abundantly. Serve right away.
NOTE: You can cook the frittata ahead. But it will solidify, when it cools, so you will have to cut it into small pieces before adding it to the pasta.