Heat charcoal, or gas grill to medium heat.
Divide the pizza dough into two equal parts, and roll them into two 10-inch pizza crusts. Place the pizza crusts on back if a baking sheet.
Lightly oil the grill.
Carefully slide the dough onto the grill and cook it for about 2 minutes. Using a tong, flip it and grill for another 2 minutes or so, until the crust is nice and cooked.
Transfer the grilled pizza crusts back onto the baking sheet.
Slice a few artichoke hearts into thin slices, and chop the rest into small pieces. (I like different cuts and sizes for a nice presentation.)
Spread pesto evenly on both pizza crusts.
Generously sprinkle a cup of mozzarella cheese on each pizza crust.
Then place artichoke pieces, and throw in roasted bell pepper strips.
Top with parmesan cheese.
Slide the assembled pizzas back on the grill and cook until the cheese is melted, 1-2 minutes. Serve immediately.