Combine warm water, yeast and sugar in large mixing bowl. Let rest until foamy (about 10 min). Add salt & Basil EVOO to yeast mixture, stir until dissolved.
Mix in flour 1 C at a time until a sticky dough forms. Turn dough out onto a well-floured surface and knead until smooth & elastic, adding more flour as needed. Form dough into a ball & brush surface w/ Basil EVOO. Return to mixing bowl & cover w/ plastic wrap and a kitchen towel; let rest in a warm, draft free area until doubled in size (about 1 hour).
While dough is rising, slice red onion & saute over medium heat in Arbosana EVOO until soft/caramelized. Remove from heat & add truffle oil. Mix well & set aside.
Punch dough down and knead a few more times. Divide into 4 small balls. Form each ball into circular pizza crust. Place each on warm (not hot) grill and let slight grill marks appear, repeat on other side (about 2-3 min per side).
Remove crusts from grill, top w/ caramelized onions (use as your sauce), then mushrooms and cheese. Return pizzas to grill, close lid & let cheese melt and start to bubble. Remove from grill, slice, serve.