In a large skillet, heat oil over medium-high. Season chicken with salt and pepper. Cook in two batches, turning occasionally, until browned and opaque throughout, 6 to 8 minutes; transfer to a plate.
To skillet, add mushrooms, asparagus, and 1 cup water; season with salt and pepper. Cook, tossing once and scraping up browned bits, until vegetables are crisp-tender, 4 to 6 minutes. Add chicken; cook, tossing, until warmed through, about 2 minutes. Serve on top of noodles or rice, if desired.