Place the turkey, garlic, chipotles, oregano, mustard, chili powder, cumin and cayenne, if desired, in a bowl and, using your hands, mix until the ingredients are evenly incorporated.
Place the mixture on a work surface and divide into 4 balls of equal size. Form each into a patty about ¾ to 1 inch thick, tossing it back and forth between your hands. Handle the patties as little as possible; do not work more than necessary.
Prepare a grill to medium-high. When the coals are glowing red, after 15 to 20 minutes, cover with the grate. After 5 minutes, use a wire brush to thoroughly clean the grate. Brush the grate with oil. When the coals are covered with pale gray ash and you can leave your hand 5 inches above the fire for 2 to 3 seconds, the coals are ready.
Sprinkle both sides of the burgers with the salt and pepper. Place the burgers on the grate and grill until well seared on both sides, about 5 minutes for medium rare or 6 minutes for medium. Transfer to buns or a serving platter and serve immediately.
Alternatively, place a cast iron skillet over high heat and when it is hot but not smoking add the burgers to the dry pan. Cook until well seared on both sides, 8 to 10 minutes for medium rare or 10 to 12 minutes for medium.