Measure out all the ingredients: ground Wagyu beef, sea salt, black pepper, burger buns, cheese slices, lettuce leaves, ripe tomatoes, red onion, gherkins, ketchup, mustard, aioli, and unsalted butter or vegetable oil.
Divide about 680 g of ground Wagyu beef into four equal portions, handling the meat as little as possible.
Shape each portion gently into a patty around 2 cm thick and slightly wider than the buns, pressing a small indentation in the centre of each to prevent bulging.
Season both sides of each patty with 1 tsp sea salt and ½ tsp freshly ground black pepper.
Heat a cast-iron frying pan or heavy-based pan over a medium–high heat for 3–4 minutes until very hot.
Add 1 tbsp unsalted butter or vegetable oil to the pan and swirl to coat.
Cook the patties without overcrowding the pan: 3–4 minutes per side for medium-rare, 5–6 minutes per side for medium, or until an internal temperature of 71 °C is reached for well done.
About 1 minute before they’re done, top each patty with a slice of cheese, then cover the pan with a lid or foil for 30–60 seconds to melt.
While the patties rest, lightly toast the buns in a dry frying pan or under the grill until golden.
Transfer the cheesy patties to a plate and rest for 5 minutes to redistribute the juices.
To assemble, spread ketchup, mustard or aioli on the buns, add the patty, then top with lettuce, tomato slices, red onion and gherkins. Serve immediately.