Prepare the dough starter: In a bowl, combine 1/2 cup slightly warm water, 1/2 cup slightly warmed milk, 1 tbsp sugar, and 1/2 tsp yeast. Stir and let it sit until foamy, about 5-10 minutes.
Make the dough: Add 2.5 cups bread flour and 1/2 tsp fine sea salt to the yeast mixture. Stir until a shaggy dough forms. Knead by hand on a non-porous surface for about 5 minutes, stretching and folding the dough.
Incorporate butter: Add 3 tbsp softened butter to the dough and knead for another 5-10 minutes until smooth and elastic. Add a touch of flour if dough is too sticky.
First rise: Place dough in a lightly oiled bowl, turning to coat. Cover with plastic wrap and let rise until doubled in size, about 1 to 1.5 hours.
Make tartar sauce: In a bowl, mix 3 tbsp mayo, 1/2 diced pickle, zest of 1/4 lemon, juice of 1/4 lemon, 1 tsp diced shallots, 1 tbsp chopped dill, kosher salt to taste, and fresh cracked pepper. Stir well and refrigerate.
Shape buns: Divide risen dough into 6 equal pieces for larger buns. Fold edges under to form tight balls. Place on parchment-lined baking tray, flatten slightly, cover loosely with plastic wrap, and let rise for 45-60 minutes.
Egg wash and bake buns: Preheat oven to 400°F (200°C). Beat one egg and brush lightly over buns. Bake for 15-20 minutes until golden brown.
Prepare batter: In a bowl, whisk together 1/4 cup all-purpose flour, 1/4 cup rice flour, 1 tsp baking powder, a big pinch of salt, and cold club soda. Add club soda gradually until batter is thin enough to coat fish.
Fry fish: Heat oil to 350°F (175°C). Coat cod fillets in batter, letting excess drip off. Fry for 5-6 minutes until golden and crispy. Remove and season immediately with salt.
Assemble sandwich: Place bottom bun, spread tartar sauce, add fried fish, top with melted American cheese, more tartar sauce, and cover with top bun. Serve immediately.