Preheat the grill to HIGH.
Prepare the dough and shape it on top of a cutting board or a flat cookie sheet. Lightly flour the surface so that the dough does not stick.
Once the grill is hot, brush the grates with olive oil.
Slide the pizza dough off the baking sheet and onto the grill.
Close the grill and grill the dough for about 2 to 3 minutes, or until browned on the bottom.
Check if the dough is browned underneath. If it’s not browned, give it another minute.
Take the dough off the grill with a pair of tongs and put it back on top of the cookie sheet or cutting board.
Turn the pizza crust over, browned-side up.
Brush with a bit of olive oil, then spread the cream cheese over the entire pizza crust.
Add 1 cup mozzarella cheese, red onions, shredded cooked chicken, and mushrooms.
Sprinkle with remaining cheese.
Put the pizza back on the grill.
Turn the heat down to medium-high, and close the lid.
Cook for about 3 to 4 minutes, or until the cheese is bubbly and melted.
Pull the pizza off the grill onto a cutting board and let it rest for a minute or so before cutting and serving.