Season the chicken with 1/2 teaspoon pepper. Lay the prosciutto on a cutting board or plate and sprinkle with the oregano. Lay each chicken breast on a slice of prosciutto and wrap the prosciutto around the chicken.
Heat the oil and garlic in a large skillet over medium heat. Add the chicken and cook until the prosciutto is crisp and the chicken is cooked through, 6 to 7 minutes per side.
Transfer the chicken and garlic to plates. If using the wine, add it to the skillet and simmer until reduced by half, 3 to 4 minutes. Spoon over the chicken.