Add green beans, sautéing until softened, but still with a crunch, about 5 minutes.
Add mushrooms continuing to sauté until browned, about 3-4 minutes. Season with salt and pepper. Mix in parsley, lemon juice and lemon zest. Set aside.
Poach eggs, one at a time: Bring a 1-quart pot of water to a simmer. Crack 1 egg into a small dish or ramekin. Holding it close to the edge of the water, slip egg into the water. Let cook for about 4-5 minutes, until whites are set. Do not stir or touch water while cooking. Carefully remove egg with a slotted spoon and drain on a paper towel.
To plate, add arugula to a large plate or bowl and top with green bean and mushrooms. Add eggs on top.