In a small bowl or liquid measuring cup, whisk together cooking wine, chicken broth, and lemon juice; set aside.
In a small bowl combine dried oregano, salt, and pepper; set aside.
Add the chopped onion to the bottom of the Crockpot insert and lay the chicken breasts on top of the onions.
Pour the cooking wine mixture over the onions and chicken.
Sprinkle half of the oregano mixture over the tops of the chicken. Flip the chicken over and sprinkle in the remaining oregano mixture over the chicken.
Evenly distribute the minced garlic over the chicken.
Put on the Crock-Pot lid and cook on HIGH for 4-6 hours or LOW for 6-8 hours.
Shred the chicken, and then use a wooden spoon to mix the shredded chicken, onions, and cooking liquid together. Turn off the Crock-Pot and let the mixture sit, with the lid on, while you make the rice.
Cook the rice as per the package instructions.
ASSEMBLE GYRO BOWLS:
Place rice in a bowl and top with chicken gyro mixture, cucumber, tomato, lettuce, feta, tzatziki sauce, pita wedges, and black pepper.