Heat half the butter in a small nonstick skillet over medium-high heat. Cook mushrooms, undisturbed, until they begin to release their liquid, about 2 minutes; season with salt and pepper, and stir. Cook until golden brown, about 2 minutes more. Transfer mushrooms to a small bowl; stir in microgreens.
Whisk eggs in a medium bowl until well combined; season with salt and pepper. Wipe skillet clean with paper towels. Melt remaining butter over medium heat, swirling to coat bottom of pan. Pour eggs into skillet, and cook, undisturbed, until edges are set slightly. With a heatproof flexible spatula, push eggs from edge toward center, tilting pan to let uncooked eggs run underneath, until omelet is just set, 1 to 2 minutes. Place mushroom filling on 1 side of omelet. Using spatula, gently fold other side of omelet over filling. Serve immediately.