Place a chicken breast inside a zip-top bag and use a meat mallet or rolling pin to pound it to an even 1/2-inch (1.25 cm) thickness. Repeat with the other breast. Return both to the bag, add the BBQ marinade, and refrigerate for at least 30 minutes or up to 4 hours.
Preheat your grill to medium-high heat (about 450°F / 232°C). Lightly oil the onion rings and place them on the grill. Cook for 3-4 minutes per side, until tender and lightly charred. Remove from the grill.
To make the sauce, finely chop the grilled onion. In a small bowl, combine the chopped onion, mayonnaise, mustard, lemon juice, and BBQ rub. Set aside or refrigerate until ready to use.
Place the bacon strips on the grill (or on a grill-safe pan). Cook for 5-8 minutes, flipping once, until crispy. Remove and set on a paper towel-lined plate. Spread softened butter on the cut sides of the buns.
Remove chicken from the marinade, letting any excess drip off. Grill the chicken for 5-7 minutes per side. The chicken is done when an instant-read thermometer inserted into the thickest part registers 165°F (74°C).
During the last minute of cooking, place a slice of pepper jack cheese on each chicken breast to melt. At the same time, place the buttered buns cut-side down on the grill for about 1 minute to toast.
Slice each chicken breast in half crosswise. To assemble, spread the charred onion mayo on the top and bottom buns. On the bottom bun, layer lettuce, a tomato slice, a piece of chicken, two slices of bacon, and a few slices of avocado. Place the top bun on and serve immediately.