Start the Traeger grill on Smoke with the lid open until the fire is established (4 to 5 minutes). If you own a pizza stone, place it on the grill grate. Set the temperature to 450F and preheat, lid closed, for at least 30 minutes.
Combine the tomato wedges, garlic, olive oil, salt, and half the basil leaves in a blender jar or the bowl of a food processor. Blend or process until you achieve a slightly chunky sauce.
Strain over a bowl to remove excess liquid. Use the liquid for soups, such as gazpacho, or freeze into ice cubes for Bloody Marys. Transfer the sauce to a mixing bowl and stir in the olive oil and salt. Set aside.
Tear two 12-inch sheets of cooking parchment from a roll. Working with one sheet at a time, lightly dampen your work surface and lay the parchment on top.
Divide the dough into two equal portions and form into balls. Dust your hands lightly with flour. Put a portion of dough in the center of the parchment paper and flatten it with the heel of your hand. Gently stretch and pull it into a rough round about 10 inches in diameter; pinch a slight rim around the edge. If the dough springs back, let it relax for a few minutes before continuing. You can also use a rolling pin lightly dusted with flour to help you achieve a thin crust.
Transfer the dough to an upside down rimmed baking sheet. Repeat with the remaining portion of dough.
Spread a few tablespoons of sauce over the first pizza. Arrange half of the mozzarella slices on top, leaving a bit of space between each. Dust lightly with half the Parmesan.
Carefully tilt the baking sheet with the pizza and parchment onto the pizza stone. If you are not using a pizza stone, put the upside down baking sheet with the pizza directly on the grill grate. Bake the pizza until the crust is nicely browned and the cheese is melted and bubbling, about 12 to 18 minutes, depending on the thickness of the crust.
Remove from the grill. Leave the pizza stone, if using, on the grill grate. Sprinkle the pizza with 1/4 cup of the remaining basil leaves and 1/2 teaspoon hot red pepper flakes, if using. Slice into wedges. Cook the remaining pizza the same way. When finished cooking, let the Traeger and pizza stone cool for several hours.
Traeger Tip: To remove grease from your pizza stone, sprinkle it with cornmeal and let it sit overnight. Brush the oil-absorbing cornmeal off in the morning.