Preheat your grill to 400°F, or use your side burner. In your Cast Iron Skillet, melt 2 tbsp. of butter and over medium-low heat, slowly caramelize the onions. Once they have become translucent, sprinkle with a pinch of sugar and continue to gently caramelize.
While the onions are going, do the same with the mushrooms. Gently caramelize them in 2 tbsp. of butter over medium-low heat. You don't need sugar for these.
Slice the chicken breasts into slices or chunks, whatever you fancy. Prepare all ingredients to top the pizza and have them at the ready for when you start cooking.
Prepare your pizza dough. Preheat the grill to medium-high, about 400 to 425°F with the pizza stone on the grill and preheating with it.
Sprinkle the pizza stone with cornmeal and place the rolled out dough onto the stone. Allow the dough to bake for 2 to 5 minutes. Flip using the pizza peel and spread the cooked side of the dough with the roasted garlic. Top with sauce, half of the cheese, the onions, mushrooms, and chicken, followed by the rest of the cheese. Bush around the edge of the crust with the garlic butter and then sprinkle with the parmesan.
Grill bake for 3 to 5 more minutes, or until all of the cheese is melty and bubbling. Turn off the grill and remove the pizza to a baking sheet or platter, allowing the pizza stone to cool on the grill.