Whisk together yogurt, olive oil, vinegar, oregano, green onion, cumin, salt, paprika, black pepper, and cayenne pepper in a large glass or ceramic bowl. Add the chicken breast halves and toss to evenly coat. Cover the bowl with plastic wrap and marinate in the refrigerator 1 hour to overnight.
Preheat an oven to 500 degrees F (260 degrees C). Grease a shallow baking dish.
Remove chicken from the marinade and shake off excess. Discard used marinade. Arrange chicken in prepared baking dish.
Bake in preheated oven until chicken is no longer pink in the center and the juices run clear, about 25 minutes, turning occasionally. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Garnish with spearmint before serving.