Season the chicken breast evenly on both sides with the dried oregano, half of the sea salt, and half of the black pepper.
Heat a non-stick skillet over medium-high heat and cook the chicken for 5-7 minutes per side until the internal temperature reaches 165°F. Remove from heat and let rest for 3 minutes before slicing into strips.
In a food processor, combine the rinsed chickpeas, tahini, lemon juice, minced garlic, and the remaining salt and pepper. Process until completely smooth, adding a splash of water if needed to reach a creamy consistency.
Slice the whole wheat pita into triangles and lightly brush with the olive oil. Toast in a toaster oven or skillet until the edges are golden and crispy.
Slice the cucumber into rounds and halve the cherry tomatoes.
Spread the fresh hummus onto a large plate or platter. Arrange the sliced chicken, toasted pita triangles, cucumber, and tomatoes around the hummus.
Garnish the entire platter with finely chopped fresh parsley and serve immediately.