Preheat the oven to 350 degrees F (175 degrees C).

Coat both sides of chicken with oregano and cayenne pepper and place on a baking sheet. Season with salt and black pepper.

Bake in the preheated oven for 15 minutes.

Meanwhile, melt butter in a nonstick skillet over medium heat. Add mushrooms, onion, Worcestershire sauce, and red wine. Cook and stir until vegetables shrink down a bit, about 5 minutes. Season with salt and black pepper. Reduce heat to medium-low to low; cook until tender, 10 to 15 minutes.

Pour mushroom mixture on top of chicken. Continue baking until chicken is no longer pink in the center and the juices run clear, about 8 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).

Sprinkle mozzarella cheese on top of the chicken and cook until cheese is bubbly, about 5 minutes more.