Halve large chicken breasts, cover with plastic wrap (to avoid splatter), and pound to even thickness using a meat mallet or frying pan—no need to flatten, just even it out. Pat dry with a paper towel. Place in a large zipper-top bag. In a bowl, whisk together 3 tablespoons lemon juice, Greek yogurt, dried oregano, olive oil, red wine vinegar, honey, and minced garlic. Season with salt and pepper. Once combined, pour over the prepared chicken. Turn to coat and massage into the breasts; refrigerate for 2–3 hours and up to 8 hours.
While chicken marinates, prepare the tzatziki sauce so it can have a chance to chill before serving. Grate a cucumber with the large holes of a grater. Line a small bowl with a few paper towels (or clean kitchen towel) and place the grated cucumber on top. Squeeze out as much moisture as you can from the cucumber (to keep the sauce from being watery). Once drained, add to a medium bowl. Add the Greek yogurt, minced garlic, red wine vinegar, 1/2 teaspoon lemon zest, 1 tablespoon lemon juice, olive oil, and salt and pepper to taste. Cover the bowl and place in the fridge until ready to serve.
Preheat grill to medium heat (400°F). Remove chicken from the marinade and discard any remaining marinade. Generously oil the preheated grill (I drench a rolled-up paper towel in vegetable oil and, holding it with tongs, rub it over the grill grates).
Add the chicken to the grill. Cook chicken for 4–5 minutes per side or until it registers 160°F (carryover heat will take it to the safe temperature of 165°F). Remove grilled chicken to a plate and cover with foil. Once everything is removed, give the grates a quick clean with the brush. Now add the pita bread. Grill pita bread for 15–30 seconds per side or until toasted and warmed through.
Assembly: In a large bowl, toss the sliced red onion, diced cucumber, and diced tomato. Season with salt and pepper (just a pinch) and toss to combine.
While chicken is resting under the foil (5 minutes), set everything out: grilled/warmed pita, sauce, and veggies. Slice chicken breasts and assemble. Start with pita bread and top with veggies and fresh parsley, add chicken, and drizzle tzatziki sauce on top. Enjoy immediately.