Dissolve the yeast in the warm water in a large bowl and let stand for 5 minutes. Stir in 3 cups flour and the salt, stirring until smooth. Stir in an additional 2 cups flour; continue adding flour (up to 1/2 cup), 1 tablespoon at a time, stirring until the dough comes away from the bowl but is still sticky.
Turn the dough out onto a lightly floured work surface, and knead with lightly floured hands. Start by slapping the dough onto the counter, pulling it toward you with one hand and pushing it away from you with the other. Fold the dough back over itself (use a bench scraper or a wide knife to help scrape dough from surface). Repeat until it's easier to handle, about 10 times. Finish kneading normally until the dough is smooth, elastic and soft, but a little tacky, about 10 minutes. Shape the dough into a ball and transfer to a lightly oiled bowl and turn to coat. Cover with plastic and let rise in a warm place until it doubles in volume, about 3 hours.
Press it with your finger to see if it's done; an indent should remain. Gently remove the dough from the bowl and divide in half. Roll each half into a ball and put on a lightly floured baking sheet. Brush lightly with olive oil, cover lightly with plastic wrap and let rest 15 minutes.
Prepare charcoal in a chimney starter (allowing the charcoal to burn until all the charcoal is covered with a thin layer of gray ash). Add to the grill, creating a two-level fire by spreading the coals out over half the grill bottom, piling them up in a mound 3 briquettes high, and leaving the other half with no coals.
Stretch and shape each ball of dough into a rough 12-by-10-inch rectangle. Brush one side with olive oil and sprinkle with salt and pepper. Turn onto the grill, oiled-side down, over direct heat. Brush the top with olive oil and sprinkle with salt and pepper. Grill until nicely charred, 1 to 2 minutes. Flip, close the lid and grill another 1 to 2 minutes.
Transfer the pizza to a baking sheet that will fit on the covered grill. Spread half of the robiola over the top. Return the pizza on the baking sheet to the grill, cover and let warm a few minutes just to melt the cheese. Transfer the pizza to a cutting board. Scatter the shaved asparagus and then the Romano shavings over the top. Season with salt and pepper. Scatter the lemon zest over the top, then the parsley leaves and drizzle with olive oil.
Repeat with the remaining dough and topping ingredients.