For the chicken and rub: Butterfly the chicken breast halves. Combine the paprika, brown gsugar, granulated garlic, granulated onion, cumin, salt, chili powder and pepper in a resealable plastic bag and shake well to mix everything together. Add the chicken breasts and shake again to coat them evenly. Set aside in the fridge for at least 30 minutes to marinate while you prepare everything else.
Preheat a grill to high heat.
For the BBQ sauce: Take the dried chiles, tear them up, discard the stems and seeds and place in a small bowl. Pour boiling water over them and let sit for 5 to 10 minutes to re-hydrate. Drain and discard the water.
In a medium saucepan, add the canola oil, onions and garlic. Sweat over medium heat until the onions are very soft and fragrant, about 8 minutes. Season with salt and pepper. Add the chiles and bourbon and simmer for 1 minute to let some of the alcohol burn off. Add the ketchup, preserves, brown sugar and molasses and simmer for 10 minutes to let the flavors come together. Puree and set aside.
Remove the chicken from the marinade and cook on the hot grill, 5 to 7 minutes per side to get some color. Remove and allow to cool. Slice on the bias into pieces.
For the pizza. Dust a bench (or pizza peel if you have one) liberally with flour. Take the ball of pizza dough and press out into a disk, working it gently from the inside out. Continue to work and stretch the dough out until you have about a 12-inch circle. The crust should be nice and thin, about 1/8-inch thick. Set on a floured surface and press gently to flatten. Spread the BBQ sauce on the pizza in a very even, thin layer. Leave a border as desired around the edges. Top with the shaved red onion, slices of BBQ chicken and slices of cheese. Dot the top with extra patches of BBQ sauce. Slide the pizza into a hot pizza oven and bake until the crust is bubbly and charred around the edges, 12 to 14 minutes. (If baking at home, set a pizza stone in an oven at 500 degrees F to warm up for 1 hour).
When done, remove from the oven and scatter the top with fresh cilantro leaves.
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