In a large skillet over medium-high heat, heat oil. Add chicken and season with dried oregano, salt, and pepper. Cook until golden and no longer pink, 8 minutes per side. Set aside to let rest 5 minutes, then thinly slice. Assemble bowls: Place cooked quinoa or brown rice into bowl and top with tomatoes, cucumber, sliced chicken, avocado, kalamata olives, feta, and dill. Make dressing: In a small bowl, combine red wine vinegar, lemon juice, and dried oregano and season generously with salt and pepper. Slowly add olive oil, whisking to combine. Drizzle with dressing and serve.