For the sauce: Roast the tomatillos, chile(s) and garlic on a rimmed baking sheet 4 inches below a very hot broiler, until blotchy black and softening (they’ll be turning from lime green to olive), about 5 minutes. Flip them over and roast the other side.
Cool, then transfer everything to a blender, including all the juice that has accumulated. Add the cilantro or parsley and 1/4 cup water, then blend to a coarse puree. Scoop into a serving dish.
Rinse the onion under cold water, then drain well to remove excess moisture. Stir into the salsa and season with salt to taste, about 1/2 teaspoon.
For the crust: In a medium bowl, mix together the flour, yeast and salt. Measure in the water and beer. Stir to combine everything into a rough-looking mass. Cover with plastic wrap and leave at room temperature for 8 to 18 hours.
After the dough has raised, heat the oven to 500°F.
Generously oil a 13 x 18 rimmed baking sheet (a half sheet pan). Use a rubber spatula to gently deflate the dough and scrape it out onto the baking sheet. Gently coax the dough into a mostly even rectangle about 9 by 13 inches.
Spread on 2/3 cup roasted tomatillo salsa (save the rest for snacking later), leaving a 1-inch border around the outside. Scatter on the cooked chorizo, sliced red onion and crumbled goat cheese. Bake in the middle of the oven until puffy and brown, about 25 minutes.
Sprinkle with queso anejo and cilantro or parsley leaves, cut in squares and serve.