Make the marinade. In a bowl, whisk 3 tablespoons olive oil, zest and juice of 1 lemon, 3 cloves minced garlic, 1.5 teaspoons dried oregano, 1 teaspoon ground cumin, 1 teaspoon smoked paprika, 1/2 teaspoon ground coriander, 1 teaspoon salt, and 1/2 teaspoon black pepper.
Layer the slow cooker. Place half the sliced red onion on the bottom.
Add the chicken. Pour the marinade over the top and scatter the remaining onion.
Cook low and slow. Cover and cook on Low for 4–5 hours or on High for 2–3 hours, until the chicken is fork-tender and reaches 165°F internally.
Shred the chicken. Transfer chicken to a cutting board and shred with two forks. Return it to the slow cooker juices and toss to coat so it soaks up flavor.
Optional broil for char. Spread shredded chicken on a foil-lined sheet pan.
Broil for 3–5 minutes, watching closely, until edges crisp slightly. Toss once and broil 2 more minutes for extra texture.
Make the tzatziki. In a bowl, combine Greek yogurt, grated and squeezed cucumber, 1 clove minced garlic, 1 tablespoon red wine vinegar, 1 tablespoon olive oil, 2 tablespoons chopped dill, a pinch of salt, and pepper. Chill until serving.
Prep the toppings. Slice tomatoes, cucumbers, and red onion.
Crumble feta. Warm pita in a dry skillet or in foil in a 300°F oven for 5–7 minutes.
Assemble the gyros. Spread tzatziki on warm pita, pile on chicken, add veggies, and finish with feta and a squeeze of lemon. Drizzle with a little olive oil if you like.
Serve. Fold and enjoy immediately while the pita is warm and the chicken is juicy.