In large bowl or bowl of stand mixer, stir yeast into warm water to dissolve. Use wooden spoon or dough hook attachment to stir in 2 cups flour, salt and olive oil until dough forms.
On lightly floured surface, knead dough by hand 10 minutes, adding just enough remaining flour until dough is smooth, soft, elastic and only slightly sticky; OR, knead in stand mixer with dough hook on medium speed 5 to 7 minutes, adding just enough remaining flour until dough is smooth, soft, elastic and only slightly sticky.
Shape dough into ball. Transfer to lightly oiled bowl; turn once to coat. Cover with plastic wrap and let rise 1 to 1½ hours until doubled.
Meanwhile, prepare the toppings: Heat truffle oil in large nonstick skillet over medium-high heat. Add mushrooms, onion, rosemary and salt and pepper to taste. Sauté 5 to 6 minutes just until mushrooms are softened. Remove from heat.
When dough is nearly risen, heat both sides of gas grill to 500°F. After 10 minutes, turn off one burner.
Punch down dough and divide in half. Cover one half and set aside.
On lightly floured surface, stretch and roll one piece of dough to 9-inch circle (no worries if it's not a perfect circle). Brush top of dough lightly with olive oil.
Transfer dough to a lightly floured (or use corn meal) pizza peel or cookie sheet with no sides. Carefully slide dough, oiled side down, onto grill over the direct heat. Grill 1 to 3 minutes with cover down, until bottom of crust is just set and char marks appear. Use tongs to check bottom of crust, checking after 1 minute and then every 30 seconds.
Brush top of dough with olive oil, then use tongs to flip crust over, transferring to the indirect heat side of grill. Top crust with half of the mushroom mixture and half of the shredded cheese. Cover and grill 2 to 3 minutes until crust is cooked through and cheese is melted.
Remove pizza from grill - use tongs to slide pizza back onto pizza peel/cookie sheet. Sprinkle top of pizza with 1 tablespoon parsley and more pepper.
Repeat steps 6-10 with second pizza dough. Serve immediately.