• 1 lb. mixed wild, shiitake, and/or crimini mushrooms, broken into pieces or sliced ¼" thick (about 4 cups)
• 3 garlic cloves, peeled, thinly sliced
• ¼ tsp. crushed red pepper flakes
• ¼ cup plus 2 Tbsp. extra-virgin olive oil, plus more for drizzling
• 1¾ tsp. (or more) kosher salt, divided
• 1 medium onion, finely chopped
• ½ tsp. freshly ground black pepper
• ½ cup dry vermouth or white wine
• 3 cups homemade chicken stock or low-sodium chicken broth, divided
• 2 oz. finely grated Parmesan (about 1 cup)
• 2 Tbsp. cold unsalted butter, cut into pieces
• ½ tsp. finely grated lemon zest
• ⅓ cup coarsely chopped parsley leaves
• Lemon wedges (for serving)