Prepare the Spicy Aioli and refrigerate until ready to use. In a large bowl, toss the thin chicken breasts in Italian Dressing to marinade at room temperature. Preheat the oven to 400℉ and start making the Oven Crisped Bacon. This will take about 25 minutes, so plenty of time to get the grill up to temperature. Preheat the grill to 400℉ over good quality oak lump charcoal with a few mesquite chunks thrown in for flavor. This will take about 20 minutes, and should be ready just about when the bacon is done cooking.
Slice the ciabatta rolls to prepare them for grilling.
Remove the bacon from the oven and set on paper towels to drain.
Toast the inside of the ciabatta rolls over the grill until they get grill marks and toasty bits. Remove them from the grill and set aside.
Prepare the chicken for grilling by sprinkling Poultry Rub on both sides of each breast. Add the Anaheim peppers and the marinated chicken to the hot grill. Place the peppers over the hottest part of the grill. turn them frequently until the skin is charred, then remove from the flame. Set the charred peppers under an upside down glass bowl to steam. This will help release the charred pepper skin when you are ready to peel them.
Continue to grill the chicken until it is done.
Add a slice of pepper jack cheese to the top of the chicken. Close the lid on the grill for a minute to melt the cheese. Carefully open the grill lid (if you open too quickly it can cause dangerous back-flash).
Remove the chicken to a platter, and set aside to rest for 10 minutes.
Carefully split and deseed the peppers. Using a paring knife, remove the charred skin (it is tough and bitter, so you want to remove it). Cut filleted peppers into sandwich sized pieces.
Assemble the sandwich. Put about 1 tbsp of Spicy Aioli onto each bun half. Add the peppers to the bottom half. Add the bacon slices, the chicken and melted cheese, sliced and salted tomatoes, leafy lettuce, and the top bun. Cut in half before serving. It is definitely a mouthful, so you can use a toothpick to keep the sandwich together.