Place turkey, garlic, chipotle, oregano, mustard, chili powder, cumin and cayenne, if using, in a bowl. Using your hands, mix until ingredients are evenly incorporated. Divide mixture into 4 balls of equal size. Form each into a patty 3/4 to 1-inch thick. Handle as little as possible.
Prepare a grill to medium-high. Brush grill grate with oil.
Sprinkle both sides of burgers with salt and pepper. Place on grate and grill until well seared on both sides, about 5 minutes for medium-rare or 6 minutes for medium.
If cooking indoors, place a cast-iron skillet over high heat. Add burgers pan coated with cooking spray. Cook until well seared on both sides, 8 to 10 minutes for medium-rare or 10 to 12 minutes for medium.
When the burgers are almost done grilling, place a slice of cheese on top of each patty. Also, place the buns, cut sides down, on the grill to toast; remove when lightly browned.
Serve the grilled patties and toasted buns along with the remaining fixings.