For a gas grill, heat 2 outside burners on high and the middle burner on medium. Clean well with brush and lightly brush grate with olive oil.
Have all your ingredients ~ sauces, cheese and toppings ~ ready to and handy before you put the dough on the grill. It goes fast.
Prepare dough. On a flat surface shape into a large round, being careful not to go too thin that it will tear, use a little flour if needed. It does not have to be perfectly round. Lightly brush or drizzle top side with olive oil. Carefully place on grill, oiled side down. Let cook a few minutes to set, get some good grill marks but be careful not to burn. It will start to bubble up a little. Drizzle a little olive oil on top. Using a spatula and tongs, flip the dough. Let cook a couple minutes. At this point, you can remove the crust, put the toppings on and then return to the grill OR you can put the toppings on while the crust is on the grill.
Working quickly, spoon on the pesto or pizza sauce as evenly as possible. Add the burrata or mozzarella slices. Add the Sopressa, leaving a small round in the middle and crack the egg into the center. Close the lid and cook another 5-6 minutes. When cheese is melted, egg is just set and crust is done, remove to cutting board.
Add the prosciutto if using, twisting pieces and placing on top of pizza. Sprinkle with a little slat and pepper, a drizzle of olive oil and fresh herbs. Cut into wedges and serve.
Notes
If using charcoal, don't use lighter fluid ~ your crust will taste like lighter fluid. Yuck!