Double-Stack Mushroom and Chicken Cheeseburgers

Ingredients

• lb. crimini mushrooms, halved if large
• tsp. Diamond Crystal or ¼ tsp. Morton kosher salt, plus more
• lb. ground chicken or turkey, preferably dark meat
• Vegetable oil (for grill and drizzling)
• small onion, trimmed, sliced crosswise into 4 rounds
• Freshly ground black pepper
• thin slices Muenster
• potato buns
• Tbsp. mayonnaise
• Tbsp. curry mustard (such as Brooklyn Delhi) or other mustard
• Bread-and-butter pickles and shredded iceberg lettuce (for serving)