Pat the chicken breast dry with a paper towel and season both sides with the dried oregano, half of the sea salt, and half of the black pepper.
Heat the olive oil in a skillet over medium-high heat. Add the chicken and cook for 5-7 minutes per side until the internal temperature reaches 165°F. Remove from heat, let rest for 5 minutes, then slice into thin strips.
While the chicken is cooking, prepare the hummus by combining the chickpeas, tahini, lemon juice, garlic, and the remaining sea salt and black pepper in a small food processor or high-speed blender.
Pulse the hummus ingredients until smooth and creamy, adding a teaspoon of water if needed to reach your desired consistency.
Finely chop the fresh parsley and fold it into the hummus for a burst of herbaceous flavor.
Slice the cucumber into rounds and lightly toast the whole wheat pita in a toaster or over an open flame until warm and slightly crisp.
Plate the sliced chicken alongside a generous dollop of the garlic herb hummus, the toasted pita, and the fresh cucumber slices for dipping.