Light a grill and oil the grate. On a baking sheet, toss the asparagus with the 2 tablespoons of olive oil and season with salt. Grill over moderate heat, turning, until the asparagus are lightly charred and crisp-tender, 3 to 4 minutes. Transfer the asparagus to plates and keep warm.

Crack the eggs into 4 small bowls. Bring a medium saucepan of water to a simmer. Add the 1/4 cup of lemon juice and season with salt. Stir the water to create a vortex. Add the eggs one at a time, stirring gently in the same direction so they do not settle. Poach over moderate heat until the whites are set and the yolks are runny, 3 minutes. Using a slotted spoon, transfer the eggs to paper towels to drain briefly.

Set the eggs on the asparagus. Drizzle with the remaining 1 tablespoon of lemon juice and sprinkle the lemon zest and cheese on top. Season generously with pepper and serve.