Preheat the oven to 500°F (if using a baking sheet) or as high as it will go (if using a baking stone/steel; place the stone/steel in the bottom third of the oven before you start preheating). Let the oven preheat for at least 30 minutes. Then, if you’re using a baking stone or steel, switch the oven to Broil on high.
Melt the butter in a medium saucepan over medium heat. When the butter is frothy, add the mushrooms, then pour in the beer. Cook the mushrooms, uncovered, for 20 to 25 minutes or until all but 1 tablespoon or so of liquid is absorbed. Remove the pan from the heat; season the mushrooms with salt and pepper, and set them aside to cool.
Stretch or roll out your dough to a 12- to 14-inch circle, then transfer it to an oiled or parchment-lined baking sheet or a lightly floured pizza peel (if using a baking stone/steel).
Drizzle the dough lightly with olive oil. Sprinkle on half of the shredded cheese, then spread the mushrooms evenly over top, leaving a ½-inch border of dough. Finally, sprinkle on the parsley and the remaining ½ cup of cheese.
Transfer the pizza to the oven and bake until the crust is golden and the cheese has browned in spots—8 to 10 minutes on the baking sheet, 6 to 8 minutes on the baking stone/steel.
Remove the pizza from the oven and let it cool for a few minutes, then slice and serve.