Make sure chicken is pounded to an even thickness before beginning. In a gallon size zipper bag, combine chicken breasts, 3 tablespoons olive oil, lemon zest and juice, garlic, oregano, dill, salt, and pepper. Marinate for at least 30 minutes to 1 hour, turning the bag occasionally.
When ready to cook, remove the chicken from the bag and discard marinade. Heat remaining olive oil in a large skillet set over medium-high heat.
Cook chicken for 4-5 minutes per side, depending on thickness, until no longer pink in the center and registers 165 degrees in the center using an instant read thermometer. Transfer the chicken to a cutting board and let rest for 5 minutes before slicing or cubing as desired.
While chicken cooks, prepare quinoa. Follow package instructions for cooking, and season with a pinch of salt for added flavor.
To assemble the bowls: divide the quinoa or rice among 4 bowls. Top with sliced chicken, tomatoes, cucumber, olives, red onion, and feta. Drizzle as desired with tzatziki sauce and a squeeze of fresh lemon juice over the top. Garnish with dill and/or chopped parsley, and serve with pita, if desired.