Hold a long, sharp knife parallel to the work surface. Slice the meat almost in half, stopping 1/4 inch from the edge. Open the sliced chicken breast like a book. Don't worry if it's a little uneven; pounding will even out the meat. Pound the meat between sheets of waxed paper using the side of a heavy cleaver or a small pan. Pound each piece until it's 1/8 inch thick.
In a bowl large enough to fit the chicken breasts, combine the oil, lime zest, cumin, coriander, soy sauce, salt, sugar, curry powder, coconut milk, cayenne, and chile. Toss the chicken in the marinade. Cover tightly and refrigerate for 1 to 2 hours.
Remove the chicken from the refrigerator; scrape off the marinade and reserve it. Heat a grill pan, cast-iron skillet, or heavy sauté pan. While the pan is heating up, put the marinade in a smaller saucepan and heat it until boiling. Reduce the heat and let it simmer for 2 minutes, stirring to prevent burning. Transfer to a sauce bowl.
Raise the heat under the pan to high, brush it with a little oil, and cook the chicken in batches. It will turn opaque and cook very quickly. Cook the first side for just 1 minute, flip the meat over, and cook the other side until the chicken is light golden, 30 seconds to 1 minute. Transfer the chicken to a warm plate; cover loosely to keep warm. Garnish with the cilantro; serve with the lime wedges and the boiled marinade as a dipping sauce.