Season chicken with salt and pepper to taste. In a large skillet with a tight-fitting lid, heat 1 1/2 teaspoons of oil over high heat; add chicken and cook until browned, 2 to 3 minutes per side; transfer to a plate and set aside.
Reduce skillet heat to medium and heat remaining oil; add mushrooms and garlic; cover and cook until mushrooms start to soften, 2 to 3 minutes; remove lid, raise skillet heat to high and cook, stirring occasionally, until mushrooms are golden.
Reduce skillet heat to medium-high and add broth and oregano; cook until mushrooms are tender and liquid is reduced, 8 to 10 minutes. Return chicken to skillet; reduce heat to low, cover and simmer until chicken is cooked through, 10 to 15 minutes. Serve chicken topped with mushrooms and drizzled with cooking liquid.