Combine sour cream, juice, zest, salt and pepper in a medium nonreactive bowl. Store, covered in refrigerator until ready to use.
Heat a lightly oiled grill pan over medium heat. Meanwhile, brush onions and mushrooms with oil and season well with salt and freshly ground black pepper.
When grill pan is heated, add onions cut side down and cook until softened, about 10 minutes. Transfer to a plate. Add mushrooms and grill until charred and soft, about 10 minutes. Serve immediately with sour cream sauce.