If the tenders are still attached to the chicken breasts, cut them off and reserve for another use. Carefully cut each chicken breast in half lengthwise to make two longer pieces, then add to a large resealable storage bag along with the lime juice, brown sugar and 1/4 cup teriyaki sauce. Seal and use your hands to massage the ingredients together. Marinate in the refrigerator for at least 1 hour and up to overnight.
Heat a large cast-iron skillet between medium and medium-high heat. Sprinkle the onion rounds liberally on both sides with salt and pepper. Add 1 tablespoon coconut oil to the skillet, swirl to coat and add the onion rounds. Use a flat spatula to press down on the onions without moving them, and cook until deeply browned on the first side, 4 to 5 minutes. Carefully flip the onions (being careful to keep the rounds together) and cook until tender and browned on the second side, 4 to 5 minutes. Transfer the onions to a large plate or baking sheet. Reserve skillet.
Whisk together the mayonnaise and remaining 2 tablespoons teriyaki sauce in a small bowl.
If using bacon, add it to a medium nonstick skillet. Turn the heat to medium and cook until crispy, flipping halfway through, about 10 minutes. Remove to a paper towel-lined plate to drain excess oil.
Remove the chicken from the marinade and lay on a large paper towel-lined plate to absorb any excess marinade. Sprinkle the chicken liberally on both sides with salt and pepper. Set aside.
Place the buns, cut-side up, on a baking sheet and toast in the oven and cook until golden brown, 5 to 10 minutes.
Add the remaining tablespoon coconut oil to the reserved cast-iron skillet and swirl to coat. Add the chicken and cook until browned on the first side, 3 to 4 minutes. Flip the chicken and cook until browned on the second side and cooked through, 2 to 3 minutes.
Turn off the heat and top each piece of chicken with 2 pieces provolone, arranging it so that the cheese overlaps and covers the entire chicken breast. Cover and let sit until the cheese is just melted, 1 to 2 minutes. Transfer the chicken to the same plate as the onions.
Spread the mayonnaise mixture on both sides of the buns. Layer each bun with a chicken breast, onion round, avocado slices, lettuce and bacon, if using.