Clean and trim the asparagus. Toss with 1 Tablespoon of olive oil and sprinkle with salt and pepper.
Heat the barbecue to medium and close the lid to retain the heat.
Add trimmed, oiled asparagus to the grill and turn after 3-4 minutes, watching carefully not to burn the tips. Cook for about 5-8 minutes, moving the more slender spears to the top rack as they become tender.
Cut the pizza dough in half. Shape and stretch the dough to make two, 10-12′′ pizzas.
Drizzle over about 1 Tablespoon of olive oil per side and rub in evenly to coat. Lay the greased pizza dough onto the pre-heated grill. Cook for about 3 minutes and flip once the underside has grill marks and is golden and firm.
Cook the other side for 3-4 minutes then turn over again to dress your pizza.
Evenly spread pesto, then dot with a few slices of slivered garlic, ricotta cheese. Layer the asparagus and sprinkle over shredded mozzarella.
Close the lid on the grill and heat the pizza to thoroughly melt the cheeses; about 2-3 minutes more.
Remove from the grill to a cutting board. Top with parmesan shavings and fresh basil leaves, if desired.