Preheat a grill. Use a sharp knife to cut off from the bottom of each asparagus stalk. These are woody, hard, and tough to chew. Discard.
Place trimmed asparagus on a large piece of aluminum foil. You might have to do this in two batches depending on the size of the asparagus bundle. Drizzle the asparagus with olive oil, Kosher salt, and a pinch of freshly ground black pepper.
Pull up the sides of the aluminum foil, then fold tightly to create and seal the asparagus into a wrapped packet.
Place each packet on a preheated grill, then cook. Remove carefully, then rest for a few minutes. See note below.
Carefully open each packet, then finish with a squeeze of fresh lemon, grated Parmigiano Reggiano cheese, and a sprinkling of flaky sea salt. Serve.