Preheat a grill pan or grill to medium-high heat.
Holding the tip and stem of each asparagus spear, snap the asparagus where it will naturally break. Discard the ends. Toss the asparagus with oil and salt to taste. Grill the asparagus until slightly charred and pliable on all sides, 2 to 3 minutes for pencil or 5 to 7 minutes for standard. Remove to a platter and reserve at room temperature. Grill each slice of bread until crispy and slightly charred on both sides. Rub each piece of bread vigorously with the raw garlic clove. Reserve with the asparagus.
Fill a bowl with cold tap water and the ice cubes. This will be the holding area for the eggs after they have been poached. Keep the bowl near the stove.
Fill a medium saucepan two-thirds of the way with water. Add the white vinegar and bring to a boil. Reduce the heat on the burner until the bubbles in the water have subsided. Gently crack and drop each egg into the water. Cook the eggs for 3 to 4 minutes. When done, the whites will be cooked through and the yolks will be warm but still liquid. Using a slotted spoon, remove the eggs from the saucepan and place in the bowl of cold water. Reserve.
To assemble the salad: Toss the arugula with half of the lemon juice, a drizzle of olive oil and a pinch of salt. Gently toss the salad until coated with the lemon and olive oil. Add more lemon juice and oil if needed. TASTE IT! The salad should be dressed and flavorful but not soggy. Divide the arugula among 4 individual plates. Arrange the asparagus on top of the arugula on each plate.
While dressing the salad, bring the saucepan of poaching water to a simmer. Gently return the eggs to the saucepan until heated through, 1 to 2 minutes. When the eggs are hot, remove them from the pan with a slotted spoon and gently blot on a tea towel to remove any excess water. Immediately place 1 egg on top of the asparagus on each plate. Sprinkle each with some salt, Parmigiano and lemon zest. Drizzle each piece of grilled bread generously with olive oil and cut in half on the bias. Lean the toast halves against each egg.