While the grill is heating prep all of your ingredients. For the pattypan squash, cut it into 1/4 inch thick slices and lightly brush them with olive oil on both sides. For the pizza sauce I selected the pastiest sauce available to prevent my pizza toppings from sliding around while they are on the grill. Otherwise, your prep is very similar to making traditional pizza.
Place the squash slices on the grill over direct high heat (450º-550º F) and grill for 4-5 minutes, with the lid closed as much as possible. You'll want to get some nice grill marks on the squash before turning, so don't move them at all.
Once they are sufficiently grilled on the bottom, flip the slices over and top each slice with a layer of sauce, parmesan, mozzarella, and pepperoni. Continue cooking with the lid closed for about 4 minutes or until the cheese is melted and just starting to bubble.
Carefully remove the pattypan pizzas from the grill using a large spatula. Allow them to sit for 5 minutes and then sprinkle with fresh basil before serving.