1. Preheat oven to 400 degrees. On a baking sheet with a rim, toss asparagus with oil. Season with salt. Bake until asparagus is lightly browned and tender, 15 to 18 minutes (timing will vary depending upon thickness of asparagus).
2. In a small saucepan, cook balsamic vinegar and sugar over medium-high heat until syrupy and reduced to 3 tablespoons, about 6 minutes.
3. Meanwhile, bring a large skillet with 2 inches of water to a simmer over medium heat. Add cider vinegar and season with salt. Break one egg at a time into a cup, then tip cup into pan. Simmer until whites are set and yolks are soft but slightly set, about 3 minutes.
4. With a slotted spatula, scoop out eggs one at a time and drain on paper towels. With a paring knife, trim edges.
5. Divide asparagus among four plates and drizzle with reduced balsamic. Top with shaved Parmesan and an egg.

Serves 4.