Move oven rack to middle position and preheat to 400 degrees F. Lightly grease a rimmed baking sheet with nonstick cooking spray and set aside.
MAKE BBQ CHICKEN MIXTURE:
While the dough rises, mix together shredded chicken, BBQ sauce, and cilantro in a small bowl until chicken is evenly coated; set aside.
ROLL OUT AND CUT THE DOUGH:
Once the dough has finished its first rise, punch the dough down and divide it into two. Roll one of the dough balls out onto a floured surface into an 11x16” rectangle.
Starting at the bottom of one of the short sides of the dough, use a pizza cutter and a wooden ruler to make straight 1-inch wide strips on either side of the dough, leaving a 3-4 inch wide section of uncut down going down the center.
FILL AND BRAID THE DOUGH:
Distribute half of the chicken mixture down the center of the dough. Sprinkle half of the onion strips over the chicken, followed by 1 cup of shredded Monterey Jack cheese and 1 cup of Mozzarella cheese.
Fold the strips of dough over the filling, and press the strip of dough into the next dough strip on the opposite side (see the recipe video for a visual on how to do this). Once you get to the end of the braid, pinch the last 2 strips together.
Carefully transfer the braid to one side of the prepared baking sheet. Then, repeat the rolling, filling, and braiding instructions above with the remaining ingredients. Once both braids are on the baking sheet, let them rest for 10 minutes.
BAKE AND SERVE:
Bake the braids for 18-20 minutes, rotating the pans once halfway through baking, until the bread is baked through and the tops are golden brown.
Move the baking sheet to a wire rack and cool for 5 minutes. Then, cut the brain and serve with chopped cilantro and extra BBQ sauce for dipping (both optional).