1: Shape the ground wagyu beef into 4 patties, each about 4 inches in diameter and about 3/4 inch thick. Work with a light touch, handling the meat as little as possible. (It helps to run your hands under cold water first.) Arrange the patties on a plate lined with plastic wrap and cover with more plastic wrap. Chill in the refrigerator for at least 1 hour.
2: In the meantime, set up your grill for indirect grilling and heat to medium-high.
3: Make the parmesan crisps: Cover a plancha with parchment paper cut to fit. Arrange the cheese in four equal mounds on the plancha. Gently flatten each mound with the back of a spoon. (For uniformity, use a round mold to shape the grated cheese into a circle.)
4: Indirect grill the crisps on the parchment-covered plancha until spread out, bubbling, and golden, 3 to 5 minutes. Let cool slightly, then transfer the crisps on the parchment paper onto a wire rack to cool. (They’ll crisp up as they cool.) Once cool, store in an airtight container if not using immediately. Leave the plancha on the grill.
5: Make the prosciutto chips: Lightly brush the prosciutto slices on both sides with olive oil. Arrange them on the hot plancha and indirect grill until lightly browned and crisp, 30 seconds to 1 minute per side. Transfer the chips to a wire rack to cool.
6: Make the special sauce: Place all the ingredients in a bowl and whisk to mix. Cover and refrigerate for up to a week.
7: When ready to grill the burgers, set your grill up for direct grilling. Place a plancha on the grill and heat to high.
8: Just before cooking, generously season the burgers on both sides with salt and pepper, turning gently.
9: Drizzle the plancha with a little olive oil. Arrange the burgers on the plancha, leaving 2 inches between each. Cook until the bottoms are darkly browned, 3 to 4 minutes. Invert the burgers, using a spatula. Smash the burgers: Place a grill press or small cast iron skillet on the burgers and flatten them. Continue grilling until both sides are crusty and browned and the burger is cooked to taste—I like medium—5 to 6 minutes in all.
10: Transfer the burgers to the wire rack and let rest for 1 to 2 minutes. (This relaxes the meat.)
11: Meanwhile, grill the cut sides of the buns on the plancha in the burger fat.
12: To assemble the burgers, spread one of the bottom buns with the sauce. Place a piece of lettuce on top. Top with the burger, tomato, pickle slices, if using, a prosciutto chip, and a parmesan crisp. Spread more sauce on the cut side of the top bun, if desired, and place on top. Repeat with the remaining burgers. Serve immediately.