Combine the yeast, sugar and warm water in a bowl. Stir to combine. Cover and stand in a warm place for 5 minutes. Stir in 2 tablespoons of the oil. Combine flour and salt in a large bowl. Add yeast mixture and mix until combined.
Turn dough onto a floured surface and knead for about 10 minutes or until the dough is firm and elastic. Cut the dough in half. Roll dough out until 5mm–thick. Slide onto a large sheet of baking paper, stand 10 minutes at room temperature.
Combine the garlic, chilli and remaining 2 tablespoons oil in a large bowl. Add the mushrooms and stir to coat. Preheat barbecue to 230°C using all burners. Barbecue the mushrooms in two batches on the plate for 3-4 minutes until light golden and tender, remove to a bowl. Scrape the barbecue plate clean.
Spread each pizza base with 2 tablespoons of the passata and spray lightly with oil. Turn off the burners under the flat plate. Place the pizza dough, still on the paper onto the hot flat plate. Close the barbecue hood and barbecue for 10 minutes until dough is light golden. Spread the remaining passata over the bases, top with mushrooms and crumble over the ricotta, close the hood and barbecue a further 5-7 minutes until ricotta is just starts to melt. Remove pizzas to a board, top with rocket, season with salt and pepper and serve.