Place fish portions between parchment paper and press to about 1/2 inch thick. This increases surface area for a crispier crust and helps the fish cook evenly.
In a bowl, whisk together flour, potato starch, baking powder, salt, and smoked paprika. Slowly pour in cold beer while whisking until a smooth, thin batter forms. The consistency should be similar to crepe batter. Keep the batter cold for best results.
Lightly crush potato chips onto a tray, keeping irregular shards for maximum crunch. Do not crush into fine crumbs.
Season the fish lightly with salt. Dredge each piece in seasoned flour, dip into the beer batter, then press into the crushed potato chips until fully coated.
Heat oil to 350°F (175°C). Fry the coated fish for 4 to 6 minutes until deep golden brown, crispy, and fully cooked through. Transfer to a wire rack and season lightly with salt if needed.
In a bowl, mix sour cream, chopped cornichons, capers, dill, black pepper, and celery seed. Add a splash of milk to loosen to your desired consistency, then season with salt to taste.
Toast brioche buns until golden brown. Spread a generous layer of tartar sauce on both the top and bottom buns.
Assemble the sandwich with the bottom bun, crispy fish, optional American cheese, and top bun with additional tartar sauce. Gently press to set.
Finish with a squeeze of fresh lemon juice over the fish just before serving.