In a large skillet, sauté mushrooms, garlic, and thyme in a little olive oil until softened and cooked through, about 5 minutes. Place mushrooms in a small bowl, leaving the liquid in the skillet, and set aside.
Preheat outdoor grill on high.
In a small ramekin, add olive oil and garlic and set aside.
While you're preheating the grill, roll out your dough on a lightly floured 9-inch pizza pan. Gently stretch it and pull it to fit the 9-inch pizza pan.
Once done, lightly brush dough with bit of the garlic olive oil. Then place it gently on the grill (you may need a helping hand for this part). Close the lid and grill for 3-5 minutes on each side (checking every so often to make sure the bottom doesn't get too charred - every grill VARIES in temperature so keep an eye on it).
After 3-5 minutes, flip the dough on the other side, close the lid and grill for another 3-5 minutes, again checking every so often to make sure it doesn't get too charred.
Once both sides have been grilled, bring the dough inside to assemble your toppings.
Brush the remaining garlic olive oil on the dough then sprinkle mozzarella cheese on top and dollops of ricotta cheese. Then sprinkle all the cooked mushrooms on top and more thyme leaves, if desired.
Return to grill but make sure the grill is off. You will just now use the residual heat to melt the cheeses. Once cheese has melted, bring back inside and slice and serve!