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
In a small saucepan, cook balsamic vinegar and sugar over medium-high heat until syrupy and reduced to 3 Tbsp, about 6 minutes.
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. With a slotted spatula, scoop out eggs one at a time and drain on paper towels
Slice mushrooms into 4 pieces each, heat olive oil in pan. Cook mushroom for 1-2 minutes (tender) and season with some salt and pepper.
Divide asparagus and eggs among four plates and drizzle with reduced balsamic.
Top with shaved Parmesan (optional)