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Season chicken with salt and pepper. In a large skillet, heat 1 Tbsp. oil over high. Add chicken, and cook until light golden brown, 2 to 3 minutes per side. Transfer to a plate; set aside.
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Add remaining tablespoon oil to skillet. Add mushrooms and season with salt. Cover; and cook over medium heat until mushrooms release their juices, 2 to 3 minutes. Remove lid. Cook over high heat, stirring occasionally, until mushrooms are golden, 4 to 5 minutes
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Add oyster sauce, and cook, stirring, until bubbly, about 1 minute. Add broth and parsley, cook over medium-high heat until mushrooms are tender and liquid is reduced, 8 to 10 minutes.
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Return chicken to skillet. Cover, reduce heat to low, and simmer until chicken is cooked through, 10 to 12 minutes. Serve chicken topped with mushrooms, and drizzled with cooking liquid. Garnish with chopped parsley.