Place flour in a shallow dish. Season chicken with salt and pepper, then coat with flour, shaking off excess.
In a large skillet, heat oil and 1 tablespoon olive oil over medium-high. In batches, cook chicken until browned and cooked through, about 3 minutes per side. Transfer to a plate and tent loosely with foil.
Reduce heat to medium, add thyme, mushrooms, and remaining 2 tablespoons olive oil, and cook until softened, 6 minutes.
Add broth and cook, stirring, until reduced by half, 3 minutes. Season with salt and pepper. Return chicken to pan along with any accumulated juices; top with parsley.
Storage: Store any leftovers in an airtight container. They will last up to 3 days.