Fill a large bowl halfway with ice water. Bring a large saucepan of water to a boil. Add the asparagus and cook until tender, 4 to 7 minutes, depending on thickness. Drain the asparagus and transfer to the ice water; drain again.
Meanwhile, in a small bowl, whisk together the egg yolk and mustard. Whisking constantly, very slowly add the oil in a steady stream. The aïoli should be quite thick. Whisk in the lemon juice and salt.
Serve the asparagus with the aïoli and lemon wedges on the side. (Because the egg in this recipe is not cooked, there is a risk of food-borne illness.)