Season chicken all over with oregano, garlic powder, salt, and black pepper. In a large skillet over medium-high heat, heat oil. Add chicken and cook, turning occasionally, until golden brown, about 6 minutes per side. Transfer to a plate.
Reduce heat to medium. Add onion and garlic and cook, stirring occasionally, until onion is softened, about 7 minutes. Add butter and stir until melted. Add rice and cook, stirring, until warmed through, about 1 minute. Pour in liquid from pepperoncini jar. Cook, stirring occasionally, until most of the liquid is evaporated, 1 to 2 minutes. Add broth, half-and-half, and red pepper. Return chicken and any accumulated juices to skillet. Bring to a boil, then reduce heat to medium-low. Cover and simmer, stirring often, until rice is tender and an instant-read thermometer inserted into thickest part of chicken registers 165°, about 15 minutes.
Transfer chicken to a plate. Stir in mozzarella and pepperoncini until cheese is melty. Return chicken to skillet before serving.
