In a large mixing bowl, whisk together the yogurt, 1/3 cup olive oil, 3 tablespoons lemon juice, crushed garlic, parsley, mint, and salt. Transfer half the mixture to another bowl, cover and refrigerate.
Transfer 1/2 cup of the marinade to another bowl, cover and refrigerate.
Add the chicken breasts to the bowl slather with the marinade, making sure to coat the chicken thoroughly.
Cover with plastic wrap and refrigerate for 2 to 3 hours or overnight.
Remove the chicken and the reserved marinade from the refrigerator.
Pre heat the grill to medium high heat. When the grill is hot, brush the grill with the 2 Tablespoons of oil.
Remove the chicken breasts from the marinade, letting the excess drip off.
Add the breasts to the grill and cook, turning often, until the breasts are evenly grilled on both sides and the internal temperature is 160 degrees.
Remove the chicken from the grill to a clean platter, cover with foil, and allow to rest for 5 to 10 minutes.
Serve with the reserved marinade and enjoy!