MARINATE CHICKEN: Prepare the chicken breasts by halving large breasts horizontally. Cover chicken breasts with plastic wrap (to avoid splatter). Using a meat mallet or the bottom of your frying pan, pound breasts to even thickness (not flattening the meat, just evening the thickness throughout). Pat dry with a paper towel. Place in a large zipper-top bag. In a bowl, whisk together the 3 tablespoons lemon juice, 1/4 cup Greek yogurt, 2 teaspoons dried oregano, 1/4 cup olive oil, 1 tablespoon red wine vinegar, 1 tablespoon honey, and 1 tablespoon minced garlic. Season with salt and pepper. Once combined, pour over the prepared chicken breasts. Turn to coat and massage into the breasts; refrigerate for 2-3 hours and up to 8 hours. I like to make the tzatziki sauce when marinating chicken because it becomes more flavorful as it sits.
TZATZIKI: While chicken marinates, prepare the sauce so it can have a chance to chill before serving. Grate a cucumber with the large holes of a cheese 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 extra moisture as you can from the cucumber (to keep the sauce from being watery). Once it's drained add to a medium-sized bowl. Add in the 1/2 cup Greek yogurt, 1 teaspoon minced garlic, 1 and 1/2 teaspoon red wine vinegar, 1/2 teaspoon lemon zest, 1 tablespoon lemon juice, 1 and 1/2 tablespoons olive oil, and salt and pepper to taste. Cover the bowl and place in the fridge until ready to serve.
GRILL: Preheat grill to medium heat (400 degrees F). Remove the chicken from the marinade and discard any remaining marinade. Prepare the preheated grill and generously oil the grill (I drench a rolled-up paper towel in vegetable oil and, holding it with tongs, rub it over the grill grates).
GRILL CONT.: Add the chicken to the grill. Cook the chicken for 4-5 minutes per side or until it registers 160 degrees F (carryover heat will take it to the safe temperature of 165 degrees 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 the pita bread for 15-30 seconds per side or until toasted and warmed through.
VEGGIES: In a large bowl, toss the sliced red onion, 2 cups diced cucumber, and 2 cups diced tomato. Season with salt and pepper (just a pinch) and toss to combine.
ASSEMBLY: While the 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 on top, and drizzle tzatziki sauce on top. Enjoy immediately.