For the chicken: Whisk together the olive brine, olive oil, oregano, anchovies, garlic, lemon zest and juice and onions in a bowl. Put the chicken in a resealable plastic bag and add the marinade. Evenly coat, press out air, seal and place in a large bowl or dish. Refrigerate for at least 2 hours and up to 8 hours.
For the salsa verde: Combine the olive oil, olives, parsley, mint, garlic, lemon zest and juice, black pepper and chile pepper if using in a bowl. Let sit for at least 30 minutes at room temperature.
Remove the chicken from the marinade. Let sit at room temperature for 30 minutes before grilling.
Prepare the grill for medium-low heat. Sprinkle the chicken lightly with salt and pepper. Place on the grill skin-side down and cook slowly until the fat is rendered and the skin begins to turn crispy and golden brown. Flip and continue grilling until the meat registers 155 degrees F on an instant read thermometer, about 20 minutes total for the smaller breast pieces and 30 for the drumsticks and thighs. Serve with the green olive salsa verde.