In a small bowl or jar combine apple cider vinegar, water, sugar and 1/4 tsp kosher salt. Stir until sugar and salt dissolve. Add the thinly sliced shallot, push down so they are submerged, cover and refrigerate for at least 20 minutes while you prepare the rest. Keeps up to 1 week.

Seared beef (Birkin patty)

Form patties

Divide the ground beef into four equal portions (about 5 oz each). Lightly form into compact patties about 3.5–4 inches wide and 3/4-inch thick, creating a slight dimple in the center of each to prevent doming. Do not overwork the meat.

Season

Season both sides of each patty with kosher salt (about 1/2 tsp per side depending on preference) and freshly ground black pepper. Brush or pat 1/4 tsp Worcestershire sauce onto one side of each patty.

Sear

Heat a heavy skillet (cast-iron preferred) over medium-high heat until very hot. Add 1 tablespoon neutral oil and swirl. Place two patties in the pan (don’t overcrowd). Sear without moving for 2–3 minutes until a deep brown crust forms. Flip, add 1/2 tablespoon unsalted butter to the pan, tilt the pan and spoon the melted butter over the patties for 30–45 seconds while the second side finishes—about 2–3 minutes for medium doneness. If making all four patties, transfer cooked patties to a warm oven (200°F / 95°C) while you cook the rest.

Add cheese

During the last 30–45 seconds of cooking, place a slice of aged white cheddar on each patty and cover the pan with a lid or plate to melt the cheese quickly.

Rest

Transfer patties to a plate and let them rest for 3–4 minutes to allow juices to redistribute.

Cheese & sauce

If using store-bought herbed garlic mayo, stir in Dijon mustard and lemon juice to brighten and thin slightly. Taste and adjust seasoning. If making quick homemade version: combine 1/2 cup mayonnaise with 1 small grated garlic clove (or 1/2 tsp garlic paste), 1 tsp chopped fresh parsley, 1 tsp Dijon, 1/2 tsp lemon juice and salt to taste.

Brioche bun and assembly

Prepare buns: split the brioche buns and lightly butter cut sides with softened butter. Toast cut sides in a skillet over medium heat or under a broiler until golden—about 1–2 minutes—being careful they don’t burn.

Assemble the Birkin: spread about 1 tablespoon of the herbed garlic mayo mixture on both top and bottom bun interiors. On the bottom bun place a small handful (about 1/2 cup) of baby arugula, then a slice of heirloom tomato (season tomato lightly with salt and pepper), then the cheddar-topped beef patty. Add 2 cornichons halved lengthwise or 2 whole, and a few quick-pickled shallot slices if using. Top with the bun crown and gently press.

Serve immediately while patties are warm. Provide extra condiments (mustard or more mayo) on the side if desired.