Preheat a charcoal or gas grill for direct, medium-high grilling. If using charcoal, push 2/3 of hot coals to 1 side of grill and spread remaining 1/3 on other side to create a multi-zone fire. If using a gas grill, reduce heat on 1 side of grill to medium.
Dust a 14-inch-wide pizza peel with flour or cornmeal; generously flour a work surface. Roll dough into a 12-inch round, adding more flour as needed to prevent sticking. Transfer dough to pizza peel; brush with oil.
Have your toppings within reach of the grill. Slide dough onto hottest part of grill; cover grill. Let cook, rotating as needed, until bottom is blistered and browned, about 2 minutes. Flip dough onto cooler side of grill so cooked side is facing up. Quickly arrange fresh mozzarella on crust, spacing evenly, then top with dollops of pesto and goat cheese. Top with artichoke hearts.
Cover grill and cook until bottom crust has browned and cheese has melted, 2 to 3 minutes, occasionally uncovering and checking bottom of crust to ensure it's not burning. If it is, reduce heat (if using a gas grill) or move to a cooler part of grill (if using charcoal). Transfer pizza to a cutting board and let stand for 2 minutes. Slice and serve.
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