If using dough stored in the refrigerator, place on a lightly floured surface, gently press into a 5-inch flat circle and cover with plastic wrap about two to three hours before you plan to bake the pizza. However, if you made your dough right away and it's already risen and come to room temperature, you're ready to skip straight to step 2.
About 45 minutes to an hour before baking, place a pizza stone in the cold oven and turn the heat to 500 degrees. If you don't have a pizza stone, you can use a large upside-down baking sheet and preheat for 30 minutes, though a pizza stone is recommended for best results.
Pour the coq au vin sauce into a microwavable bowl or measuring cup. Heat until the mixture is warm and stir.
In a skillet over medium heat, melt two tablespoons of butter. Whisk in flour and cook for about three minutes. Continue to whisk constantly. Pour in coq au vin sauce and whisk until smooth and thickened. Remove from heat.
Also in a skillet, melt the remaining two tablespoons of butter. Stir in the mushrooms to coat with butter and cook until golden, about four to five minutes. Remove from heat.
Hand stretch the dough using the bed of your fingers rather than your finger tips to pat the dough outward toward the edge. Don’t pat in the center of the dough, or you’ll get limp pizza. Really focus on staying outside of the center and work toward the edge. After you work the dough, grab the edge of the pizza and lift it in the air. Allow it to quickly hang down and keep rotating the pizza crust through your fingers to allow the rest of the pizza to have a chance to hang. Do not use a rolling pin to shape and flatten. The dough is stretched when it’s 10 inches in diameter. If any holes form, pinch the dough back together.
Coat a pizza peel or the back of a parchment paper-covered baking sheet with cornmeal and place the hand-stretched dough on top. Adjust the shape if necessary.
Spread ¼ cup of coq au vin sauce on the dough, leaving an edge for the crust. Use the remaining sauce to make additional pizzas or for a tasty dipping sauce for your crust. Top with mozzarella and then add the mushrooms, chicken and red onions. Whisk together the egg yolk and water and brush on the crust. Transfer to the hot pizza stone.
If using parchment paper to transfer the pizza, bake for two minutes, remove the parchment paper and then continue to bake for six to eight minutes. If using a pizza peel with no parchment paper to transfer the pizza directly to the stone, bake for eight to 10 minutes.
Remove from oven and let the pizza set for about five minutes. Cut into slices and top with fresh parsley and Parmesan if desired. Enjoy!