Mince the garlic cloves finely and slice the green onion into thin rings. Chop the prawns into 1/2-inch pieces—I like to keep some texture in the mixture rather than making it completely smooth, which gives the burgers a more interesting bite. In a medium bowl, combine the chopped prawns with the minced garlic, green onion, salt, black pepper, smoked paprika, and lemon zest. Mix gently but thoroughly until the seasonings are evenly distributed throughout.
Crack the egg into a shallow bowl and whisk it well with a fork. Place the all-purpose flour in a second shallow bowl and the panko breadcrumbs in a third bowl. Arrange these three bowls in a row—this is your breading station. Having everything organized before you start coating will make the process quick and prevent the prawn mixture from getting too warm, which can affect texture.
Add the beaten egg to the prawn mixture from Step 1 and mix gently until just combined—the egg acts as a binder to hold everything together. Divide the mixture into 4 equal portions and shape each into a tight ball. Working with one ball at a time, coat it lightly in flour (shaking off excess), dip it in the egg wash, then press it firmly into the panko, making sure all sides are covered. Gently flatten each breaded ball into a patty about 3/4-inch thick. I find that pressing the patties between two pieces of parchment paper helps them hold their shape during cooking without compacting them too much.
Heat the olive oil in a large skillet over medium-high heat (around 375°F if using a thermometer) until it shimmers and a small piece of panko sizzles immediately when dropped in. Carefully place the patties in the hot oil, being careful not to overcrowd the pan—they should have space around them for even browning. Cook for 2 minutes on the first side until golden brown, then flip and cook for another 2 minutes on the second side. The exterior should be crispy and golden while the interior remains tender and moist.
While the patties are cooking, slice the brioche buns and lightly butter the cut sides. Toast them in a separate skillet or toaster until they're golden brown and slightly crispy—this prevents them from getting soggy when the sauce and patty are added. Shred the lettuce into 1/4-inch strips if not already done. Once the patties finish cooking, remove them to a plate lined with paper towels to drain any excess oil.
Spread bang bang sauce on the bottom half of each toasted bun. Layer in order: the cooked shrimp patty from Step 4, a few gherkins, and a handful of shredded lettuce from Step 5. Top with the other half of the bun and serve immediately while the patties are still warm and crispy.