Preheat oven to 450°F and line sheet pan with a wire rack.
Set up three-stage breading process with three shallow bowls. In the first, spread out flour. In the second, whisk buttermilk and egg until smooth. In the third, toss bread crumbs, Parmigiano-Reggiano, oil, garlic powder, salt, and pepper.
First, roll each asparagus spear in flour. Then, dip in buttermilk mixture, letting excess drip off. Next, coat with bread crumb mixture. Place breaded asparagus on wire rack. Bake until golden brown, 15 minutes,
While they're baking, mince and mash garlic to a paste with a pinch of salt using a large heavy knife.
Whisk together yolk, lemon juice, and mustard in a bowl. Combine oils and add a few drops at a time to yolk mixture, whisking constantly, until oil is incorporated and mixture is emulsified.
Whisk in garlic paste and season with salt and pepper to taste. Chill until ready to use.