As the mushroom mixture is cool, preheat the oven to 400°F (205°C) with a rack in the center position. Line a half sheet pan with parchment paper and set aside.
Remove the dough from the fridge and lightly dust with flour. If it is too cold, allow it to sit at room temprature for 5 to 10 minutes before proceeding. On a floured work surface, roll into a 12-inch round. Transfer to the lined baking sheet. Note: If the dough warms too quickly or becomes sticky, chill in the refrigerator for 10 to 15 minutes before proceeding.
Crumble the blue cheese (or goat cheese) into a large bowl, add the cooled mushroom mixture (discard the thyme and rosemary sprigs), and stir together gently.
Spoon the mixture into the center of the dough, spreading it in a thick even layer, leaving a 1-1/2 inch border on all sides. Fold and pleat the edges of the dough border over the mushrooms – leaving the center exposed. Lightly brush the edges of the dough with egg wash.
Bake for 40 to 50 minutes, rotating the pan halfway, or until the crust is golden brown in color. Place on cooling rack and allow to cool for 5 to 10 minutes before slicing and serving. Garnish with fresh thyme leaves, as desired. Serve hot, warm, or at room temperature.