In a large pot of boiling salted water, cook pasta according to package instructions, adding the asparagus in the last minute of cooking. Reserve 1 cup cooking liquid; drain pasta and asparagus and return to pot along with butter and Parmesan. Toss until butter is melted, adding enough pasta water to create a thin sauce that coats pasta.
While pasta is cooking, in a large straight-sided skillet, heat 2 inches water over medium until a few bubbles rise to the top. Crack each egg into a small bowl and gently pour into skillet. Cook until whites are set and yolks are runny, 4 minutes. With a slotted spoon, transfer eggs to a parchment-lined baking sheet. (See adaptation above.)
To serve, divide pasta among four bowls, top each serving with an egg, and season with salt and pepper. Sprinkle with Parmesan if desired.
Yields 1 cup
Place in the top of a double boiler or in a large stainless-steel bowl set up as a double boiler:
3 large egg yolks
1 1/2 tablespoons cold water
Off the heat, whisk the egg mixture until it becomes light and frothy. Place the top of the double boiler or the bowl over- not in- barely simmering water and continue to whisk until the eggs are thickened, 2-4 minutes, being careful not to let the eggs get too hot (or they will scramble). Remove the pan or bowl from over the water and whisk to slightly cool mixture.
Whisking constantly, very slowly add 1/2 cup warm (not hot) clarified butter, 1069
1 to 3 teaspoons fresh lemon juice
Salt and ground white pepper to taste
If the sauce is too thick, whisk in a few drops of warm water. Serve immediately or keep the sauce warm for up to 30 minutes by placing bowl in warm water.