Bring the pizza dough to room temperature, about 2 hours on the counter.
Stretch out both pizza dough balls into 10-inch circles on a lightly floured counter. (Watch my video on how to stretch pizza dough.)
Heat the grill over high heat for at least 10 minutes.
When ready to grill, clean and oil the grates. I soak veg. oil into a paper towel and run it all over the grates using a pair of kitchen tongs.
Reduce the heat to medium high. Then carefully place each crust directly on the grill, cover and cook for about 2 minutes. Using a kitchen tong, flip it and grill for another 2 minutes or so, until the crust is nice and cooked.
Transfer the grilled pizza crusts onto a baking sheet.
Spread pesto evenly on both pizza crusts. And generously sprinkle a cup of mozzarella cheese on each pizza. Then place artichoke pieces, and throw in roasted bell pepper strips. Finally sprinkle parmesan cheese on top.
Slide the assembled pizzas back on the grill and cook until the cheese is melted, 1-2 minutes. Serve immediately.