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Make the pickle mustard sauce by combining the mayonnaise, mustard and cornichons in a bowl. Chill until needed.
Heat the barbecue and wait until the coals turn ashen. Alternatively, heat a griddle pan over a high heat. Cook the sausages for 7-8 mins, turning regularly until evenly browned and cooked through, then push the sausages together on the grill or in the pan and scatter with the grated mozzarella. Cook undisturbed for 2 mins more, or until the cheese has melted.
Split the hot dog buns open and stuff each one with a cheese-topped sausage. Drizzle over the pickle mustard sauce and top with crispy onions and ketchup, if you like.