Combine warm water, yeast and sugar in a large mixing bowl. Let rest until foamy (about 10 minutes). Add sea salt and Primo Basil Olive Oil to yeast mixture and stir until salt is dissolved.
Mix in flour one cup at a time until a sticky dough forms. Turn dough out onto a well-floured surface and knead until smooth and elastic, adding more flour as needed. Form dough into a ball and brush the surface with Primo Basil Olive Oil. Return to mixing bowl, cover bowl with plastic wrap and a kitchen towel and let rest in a warm, draft-free area until double in size (about an hour).
While dough is rising, slice red onion and sauté over medium heat in Primo Corbrancosa Extra Virgin Olive Oil until onions are very soft (caramelized). Remove from heat and add Primo White Truffle Olive Oil. Mix well and set aside.
Punch down dough, and then knead the dough a few more times. Divide the dough into four small balls. Form each ball into round pizza crust. Place each pizza crust on a warm grill (not hot). Let slight grill marks form and turn over and repeat on other side (about 2-3 minutes on each side).
Remove crusts from grill and top with caramelized onions (use as your sauce), then mushrooms and then cheese. Return pizza to grill, close lid and let cheese melt and start to bubble. Remove from grill. Layer arugula on top and drizzle Primo Fig Balsamic over the pizza. Cut and serve.