Heat gas or charcoal grill for indirect cooking as directed by manufacturer.
In 8-inch skillet, heat 3 teaspoons of the oil over medium-high heat. Cook mushrooms in oil 4 to 6 minutes, stirring frequently, until liquid is absorbed; season with salt and pepper, if desired. Set aside.
Cut 18x12-inch sheet of heavy-duty foil; spray with cooking spray. Unroll dough; place on sprayed foil. Press out dough with hands to form 13x9-inch rectangle.
Brush dough with remaining 1 teaspoon oil.
Turn dough upside down onto grill rack and peel off foil. For two-burner gas grill, place on unheated side. For one-burner gas grill, place on grill over low heat. For charcoal grill, place over drip pan placed next to medium coals. Cook 3 to 5 minutes or until bottom of dough is golden brown. Turn dough; grill 1 to 2 minutes longer or until bottom is set.
Carefully remove crust from grill and place on foil.
Spread crust with pesto; top with mushrooms and cheese. Return pizza on foil to grill heated to medium. Cook 3 to 5 minutes or until crust is thoroughly cooked and cheese is melted. Top with basil, if desired.
