Brush a large glass bowl lightly with olive oil. Place the dough ball in the prepared bowl, one that is about twice the size of the dough. Turn the dough in the bowl to coat with the oil. Cover with a clean towel and place in a warm area until doubled in size, about one hour.

Turn the dough out onto a lightly floured surface. Using the heel of your hand, gently press and stretch the dough out toward the edges. For a thin crust, stretch the dough to 1/4-inch or less thickness. Using a rolling pin to smooth it out helps with the process. If the dough is stubborn and wants to "shrink back" as you work, just give it a five minute rest before continuing.

Preheat a gas grill to medium-high heat.

When you are satisfied with the results, brush the entire surface of the dough lightly with olive oil. Season lightly with coarse salt.  Gently lift one end and fold the dough in half.  Again, brush the surface lightly with olive oil.  Lift edge of long end and again fold in half, brushing once again with olive oil.  Place oiled side of folded dough down on tray and oil again.  Let the folded dough rest for 10 minutes.  The dough is now ready for transport to the grill.

Unfold the dough and cautiously place it on the preheated grill grates.  You may need to quickly stretch the edges to re-establish the desired shape.  Close grill cover.  After a minute or so, lift the edge of the dough to check on the progress.  Once dough starts to "bubble" and gill marks are noticeable, flip the dough using metal tongs or a pizza turner.  Cover grill and after one minute, lift the dough to check underneath.  Once the char marks are apparent, the crust is ready to be transferred to a tray.  

If not using right away, allow crust to cool completely before placing in plastic bag for storage.

Thoroughly wash broccoli rabe.  Have a large bowl of ice water ready.  Bring a large pot of salted water to boil. Add broccoli rabe and blanch for 60 seconds.. With tongs, remove broccoli rabe and place immediately in ice water.  Once cooled, drain in a colander.  Lay in a single layer on paper towels to dry thoroughly.  Cut in 1" to 1 1/2" pieces.

Heat olive oil in skillet over medium heat. Add minced garlic & red pepper flakes. Sauté for about 30 seconds. Add broccoli rabe and 1/2 teaspoon salt. Sauté for one minute, tossing continuously with tongs.   Remove from heat and set aside.

Clean mushrooms by wiping top surface with a dampened paper towel.  Flip mushrooms over and remove underneath gills by scraping with the edge of a spoon.

Cut mushrooms into generous slices, about 1/2" thick.  Heat olive oil in skillet over medium heat.  Add mushrooms in a single layer.  Let them cook for about 3 to 4 minutes without moving them.  Once browned, flip mushrooms to continue cooking on other side, an additional 2 to 3 minutes.  Season with salt and pepper.  Set aside.

Place pizza stone, if using, on rack positioned in the bottom third of oven.  Preheat oven to 500°F.

Spread sliced tomato between layers of paper towels to absorb excess moisture.

Lightly brush edges of pizza crust with olive oil.  Evenly spread marinara sauce to cover entire surface, within a half inch of edge.  Place tomato slices, evenly spaced, over sauce.  Top with broccoli rabe pieces.

Carefully place mozzarella over broccoli rabe, allowing for even spacing of about an inch between slices.  Next, add the mushroom slices.

Slide pizza onto preheated stone.  Set the timer for 9 minutes.  Remove from oven when ready.  Cut into eight slices.  Top with fresh basil and freshly grated parmesan cheese.  Serve immediately.