These gyros are so balanced and filling that I usually serve them with a simple fruit salad, or just an individual fruit. A real food meal just doesn’t get any easier than that.
In a shallow and wide container, or a gallon-size Ziploc bag, whisk (or shake) the marinade ingredients together: olive oil, lemon juice, red wine vinegar, dried oregano, minced garlic, salt and pepper.
Place the chicken in the marinade, and toss the chicken around so it's covered. Marinate the breasts in the fridge for at least 30 minutes, up to 12 hours. The chicken will appear to be "broken down" if you choose to marinate for 12 hours due to the acid, but the taste and texture don't reflect this once cooked.
Heat the grill, or you can cook the chicken in a skillet or grill pan. If you're using a grill, you'll want the temperature around 400F--anything hotter will quickly dry out the breasts. Grease the grill or pan with a small amount of extra olive oil.
Remove the chicken from the marinade (toss the extra marinade in the trash), and cook the chicken directly over the heat until it's brown on the side that's facing the heat. Once brown, flip the chicken and continue to cook on the other side until brown. The time it takes for your breasts to cook will vary depending on the size of the breasts. I usually cook the chicken breasts for about 10 minutes on each side. The breasts are done when a meat thermometer, inserted into the breasts, registers at 160F.
Let the breasts rest for 5-10 minutes before slicing. Slice the breasts. While the grill or pan are hot, warm the pitas one by one for just a minute on each side. This is optional, but so delicious!
Assemble the gyros by topping a flat pita with chicken, your desired toppings, and homemade tzatziki sauce.