Step 1Heat 1 tablespoon oil in a small skillet over medium heat. Add the green garlic and a large pinch of salt. Cook, stirring occasionally, until bright green and tender but not brown, about 3 minutes. Remove from the heat and cool.
Step 2Transfer the cooled green garlic to a blender, along with the egg and lemon juice. Puree until smooth. With the machine running, add the remaining half cup oil in a slow, steady stream until the mixture is thick like mayonnaise. Season with salt.
Step 3Make Ahead: The mayonnaise can be refrigerated for up to 1 week.
Step 11 Whisk the vinegar, lemon juice, mustard, pepper, and a pinch of salt in a large bowl. While whisking, add the oil in a slow, steady stream and continue whisking until the vinaigrette emulsifies. Season with salt.
Step 2Make Ahead: The vinaigrette can be refrigerated for up to 1 week.
Step 11 Bring 8 cups water to a boil in a large saucepan over high heat and add 2 tablespoons salt. Add the asparagus and cook, turning occasionally, until bright green and just tender, 3 to 6 minutes. Drain and immediately spread on a paper towel-lined baking sheet in a single layer. Refrigerate uncovered until chilled.
Step 22 Meanwhile, put the eggs in a small saucepan and add enough cold water to cover by 1 inch. Bring to a boil over high heat, then cover and remove from the heat. Let stand for 9 minutes. Drain and run under cold water until cool. Peel the eggs and coarsely chop, then transfer to a bowl with the vinaigrette. Add the breadcrumbs and parsley and fold gently until evenly mixed.
Step 33 Pull the paper towels out from under the asparagus and discard. Drizzle the olive oil over the asparagus and sprinkle with salt. Turn the spears with your hands to evenly coat.
Step 44 Spread the mayonnaise all over a serving platter and top with the asparagus, then the egg salad. Serve immediately.