Prepare a Grill Fire: We are using lump charcoal with hickory chunks for smoke flavor. Start the fire and place a pizza stone directly onto the grill grate. We like to cook pizza at between 350° and 400°.
Prepare the Pizza Sauce: In a bowl, mix together the tomato sauce and basil pesto.
Roll Out the Dough: On a lightly floured surface, roll out your pizza dough to your desired thickness to about an 10 to 12-inch circle. Fold the edges over to create the crust. Place on a piece of parchment paper on a flat baking sheet or pizza peel.
Assemble the Pizza: Brush the pizza dough with olive oil in a thin coating, covering the bottom and edges. Spread a thin layer of the sauce over the dough. Top with mozzarella cheese, followed by ham chunks, pineapple chunks and red onion. Add a bit more shredded cheese.
Grill the Pizza: Slide pizza onto the hot pizza stone with the parchment paper. Close the lid and cook for about 5 minutes. Carefully slide the parchment paper out from under the pizza. Use a pizza steel or large spatula to turn the pizza a quarter turn and continue to cook with the grill closed, turning every five minutes. Cook until the crust is golden and the cheese is bubbly.
Make the Garlic Butter: In a small skillet melt butter over medium heat, add sliced garlic cloves and cook, stirring often about 2 minutes, set aside
Serve and Enjoy: Once pizza is done, carefully remove the pizza from the grill onto a tray or cutting board. Brush garlic butter over the edges of the crust and let it rest for a few minutes, then slice and serve.