Cook the bulgur according to package instructions until tender, then fluff with a fork and let it cool completely.
Season the chicken breast with half of the sea salt and black pepper, then grill over medium-high heat until cooked through.
Slice the grilled chicken into thin, bite-sized strips.
In a small glass jar or bowl, whisk together the extra virgin olive oil, fresh lemon juice, dried oregano, and the remaining salt and pepper.
In a large mixing bowl, combine the cooled bulgur, diced cucumber, halved cherry tomatoes, and chopped fresh parsley.
Drizzle the lemon-herb dressing over the bulgur mixture and toss thoroughly to ensure every grain is coated.
Transfer the salad to a serving plate, top with the sliced chicken strips, and sprinkle with crumbled feta cheese.