Place a steamer insert into a saucepan and fill with water to just below the bottom of the steamer. Bring water to a boil. Add broccoli, cover, and steam until tender, 2 to 6 minutes. Drain.Advertisement
Heat olive oil in a large saucepan over medium heat. Cook and stir chicken until no longer pink in the center, about 5 minutes. Add onion; cook and stir until translucent, about 5 minutes. Reduce heat to medium.
Stir broccoli, cream of mushroom soup, and mushrooms into the saucepan. Cook until heated through, about 5 minutes. Add milk and soy sauce; cook until flavors combine, 8 to 10 minutes.