1. Prepare the asparagus by snapping off the tough woody ends naturally; peeling stalks is optional but recommended for thicker spears.
2. Preheat the grill or grill pan to medium-high heat (about 350°F to 400°F) until smoking or very hot to achieve good char marks.
3. Toss asparagus with about 1 tablespoon olive oil, coating well, then season generously with salt and freshly ground black pepper. For pan grilling, omit oil in the initial step and season after cooking.
4. Arrange asparagus spears perpendicular to the grill grates to prevent slipping and grill with the lid closed, turning once or twice, until tender and browned in spots, about 5 to 10 minutes (4 to 6 minutes for thinner spears).
5. Meanwhile, mix lemon juice with minced shallots, stir in remaining olive oil to balance the lemon’s sharpness, then season with salt and pepper and add parsley if using.
6. Serve asparagus hot or at room temperature, drizzling with the lemon shallot dressing and lemon wedges if desired. This dish pairs well with grilled fish or salmon.