Heat oil in skillet over medium heat. Add asparagus and cook 4-6 minutes, turning occasionally, until tender but still slightly crisp. Remove from heat. Let cool slightly before wrapping each asparagus with a piece of prosciutto.
Meanwhile, bring a medium pot of water to a gentle simmer (not a rolling boil). Add vinegar.
Crack 1 egg into a small bowl. Gently stir water to create a swirl and carefully slide egg into pot. Repeat with remaining eggs. Cook 3-4 minutes for a soft, runny yolk. Remove with a slotted spoon and place on a paper towel to drain.
Place toasted bread on a plate. Top with asparagus, avocado, poached eggs and Parmesan.
For firmer yolks, cook eggs closer to 4–5 minutes.
Asparagus can also be roasted at 400°F for 10–12 minutes, if preferred.