Pound chicken breasts to ¾-inch even thickness. Mix all spices and rub evenly over both sides of each breast.
Preheat oven to 400°F (200°C).
Heat oil in a large oven-safe skillet over medium-high heat until shimmering. Sear chicken for 4 to 5 minutes undisturbed until deeply golden. Flip and sear 3 minutes more. Transfer to a plate — chicken will not be cooked through yet.
In the same skillet, melt butter with oil over medium heat. Cook onions for 8 to 10 minutes until softened and golden. Add mushrooms, garlic, salt, pepper, and thyme. Cook on medium-high for 5 to 6 minutes until mushrooms are caramelized and their liquid has fully evaporated. Add soy sauce and stir 30 seconds.
Return chicken to the skillet, nestle into the mushroom mixture. Spoon mushrooms and onions over each breast. Divide grated cheese evenly over the top of each chicken breast.
Transfer to the oven and bake for 8 to 12 minutes until cheese is melted and golden and chicken registers 165°F (74°C).
Rest 3 minutes. Scatter fresh parsley over the skillet. Serve with mashed potatoes, rice, or vegetables.