Preheat an oven to 400 degrees F. In a small bowl, mix together the cheese and mayonnaise. Season the chicken breasts with salt and pepper and arrange them on a baking sheet. Evenly divide the cheese mixture between the chicken, spreading it out to cover the surface of the meat. Bake the chicken until the cheese is golden brown and chicken is cooked through, 15 to 17 minutes.
While the chicken is baking, in a large serving bowl whisk together the olive oil and vinegar; season with salt and pepper. Toss the vegetables and oregano in the dressing. To serve, slice the chicken and arrange it over the chopped salad.