Set a large nonstick skillet over medium heat. Add 3 tablespoons of the olive oil and heat to shimmering, then add the mushrooms and rice. Cook, stirring occasionally, until the mushrooms and rice turn crisp and light golden brown, about 3 minutes. Add the peas, asparagus, and coriander and season with a pinch of salt and a twist of pepper. Continue cooking until the vegetables soften, about 2 minutes. Remove from the heat, stir in the parsley, and divide the rice between two plates.
Return the skillet to medium heat. Add the remaining 1 tablespoon olive oil and heat to shimmering. Carefully crack the eggs into the pan and cook until the whites begin to brown around the edges, about 2 minutes. Season with a pinch of salt and a twist of pepper. Remove from the heat, add 1 tablespoon water, cover, and let steam for 30 seconds for runny yolks, 1 minute for medium-set yolks, and 1½ minutes for fully set yolks.