Whisk the ketchup, lime juice, apple cider vinegar, honey, molasses, Worcestershire sauce, Dijon mustard, liquid smoke, smoked paprika, cayenne pepper, salt, pepper, and water in a small pot.
Add the spring of rosemary, 1/2 of the red onions to the sauce,and heat it over medium-high heat until it comes to a boil.
Decrease the heat to medium-low; cook it for 10 minutes, and allow the BBQ sauce to cool completely.
Add 1/2 of the BBQ sauce to a gallon-sized resealable Ziplock bag, then add the chicken breast.
Massage the rosemary-infused BBQ sauce into the chicken and let it marinate for a minimum of 1 hour or overnight in the fridge.
Remove the chicken from the marinade and program your grill to medium-high heat.
Poke holes into a sheet of non-stick foil, then arrange it on the grill with the non-stick cooking surface facing upwards.
Arrange the chicken breast, the remaining onions, and lime slices on the foil and cook it for 12-15minutes, basting it every so often with the remaining BBQ sauce.