Braise chicken: Season meat liberally with salt and pepper. On stove top in a saute pan over high heat, warm oil and caramelize shallots and garlic. Add chicken and brown on all sides. Once exterior is browned, pour 1 can of beer, turn down heat to low-medium, and cover the pan. Flip and rotate the chicken intermittently, removing meat and onions from pan when fork tender and liquid absorbed. Slice across the grain and set aside to cool.
Preheat grill: 20 minutes in advance of grilling, preheat the grill to 450F using the 2-zone cooking method. Apply 1 perforated tin foil packet wood chip packet in the back corner of the direct heat grilling zone (see Notes Section below).
Pizza Prep: Roll out dough on a floured surface to ¼-inch thickness. Brush dough with oil and dust with cornmeal. Set aside. Prepare toppings as instructed above and organize into separate containers for quick, easy access during grilling. Set aside.
Grilling the Pizza: Gently lay rolled dough over the direct heat grill grates searing the first side (with grill lid open), for 3-5 minutes or until dough rises and edges begin to brown. Flip dough over and onto the cooler indirect heat grilling zone. Immediately brush with oil and being dressing with cheese, sliced chicken and caramelized shallots, chorizo, and figs. Close the grill lid and allow the pizza to bake until cheese bubbles and dough is cooked through. Remove pizza from grill and place on cooling rack to rest for 3-5 minutes. Season to taste with salt and fresh ground peppercorn, drizzle with balsamic reduction, and finish with shaved Parmesan, fresh leaves of basil and thyme. Slice and serve.