Pat chicken breasts dry with paper towels. If breasts are unevenly thick, pound between plastic wrap to achieve uniform 3/4-inch thickness.
In a glass bowl, whisk Greek yogurt until smooth. Add olive oil, microplaned garlic, lemon zest, lemon juice, oregano, salt, pepper, cumin, and cayenne (if using). Whisk until fully incorporated.
Add chicken breasts to the marinade, turning repeatedly to ensure complete coverage. Massage the marinade into the chicken, focusing on natural seams in the meat.
Cover the bowl with plastic wrap or transfer to an airtight container. Refrigerate for 4-8 hours (optimal) or up to 24 hours maximum.
Remove chicken from refrigeration 30 minutes prior to cooking. Gently remove excess marinade with your fingers, leaving a thin coating.
Heat a heavy-bottomed skillet over medium-high heat with 1 tablespoon olive oil. Cook chicken breasts for 5-7 minutes per side until an instant-read thermometer registers 165°F at the thickest point.
Transfer cooked chicken to a clean plate and tent loosely with aluminum foil. Rest for 5-7 minutes before serving to allow for moisture redistribution.