Bring a large pot of water to boil. Add trimmed asparagus and boil for 8 minutes, until tender. Drain water and run asparagus under cold water to stop cooking.
Toss asparagus in olive oil. Arrange asparagus on two plates and sprinkle with even amount of salt, bread crumbs, and Parmesan.
Bring a small pot of water to boil. Add vinegar. Crack in eggs and stir vigorously. Cook for 2-2½ minutes. Using a slotted spoon, transfer one egg to each plate. Top with parsley and freshly ground black pepper.