Preheat the grill: Heat a gas or charcoal grill to medium-high (about 400–450°F / 205–230°C). Brush and lightly oil the grates to help prevent sticking.
Make the kale pesto: In a blender or food processor, combine the kale (for pesto), basil (for pesto), nuts, lemon juice, garlic (for pesto), salt, and pepper. With the motor running, drizzle in the 1/4 cup olive oil until you have a spoonable pesto. Set aside.
Shape the dough: On a lightly floured surface, stretch or roll the dough into about a 12-inch round (or your preferred shape). Brush the top surface with 2 tablespoons olive oil (stir in the minced garlic if using).
Grill the first side: Place the dough oiled-side down directly on the grill grates. Close the lid and cook 2–3 minutes, until the bottom is lightly charred with grill marks and the dough releases easily.
Flip and top: Brush the now-upward side with a little olive oil. Using tongs (and a peel if helpful), flip the crust. Quickly spread about 1/2–3/4 cup kale pesto over the grilled side (use more or less to taste). Scatter the torn kale (topping), add sliced tomatoes, then distribute the torn mozzarella. Season lightly with a pinch of salt and optional red pepper flakes.
Finish grilling: Close the lid and cook 3–4 minutes more, until the cheese is melted and the bottom is crisp and browned. If it’s browning too quickly, slide the pizza to a cooler spot on the grill.
Rest and serve: Transfer to a cutting board, rest 1 minute, then top with fresh basil leaves. Slice and serve right away.