Set a wire rack over a sheet pan. Cut the haddock into bun sized portions, about 2 ounces. If a piece is very thick, butterfly it by slicing almost all the way through horizontally, then open it like a book and gently press to flatten. See notes.
Lightly sprinkle both sides of the fish filets with 1/2 teaspoon of salt and let it sit for 10 minutes on the wire rack. This will extract any excess moisture.
While the fish rests, set up your breading station with three bowls. Add flour to the first dish. Add the eggs to the second dish, and whisk until beaten.
In the third dish, combine the panko breadcrumbs, parmesan cheese, and the remaining ¼ teaspoon of salt. Pour the melted butter over the breadcrumb mixture and stir until the crumbs are evenly coated.
Once the fish has rested, pat each filet dry with a paper towel. Dredge each filet in the flour, shaking off any excess.
Next, Then coat the fish completely in the beaten egg. Then add the eggy fish to the breadcrumb mixture. Coat and press gently to get the breadcrumbs to stick to the fish. Shake off any loose crumbs and set the breaded fish on a plate or back on the wire rack. Repeat the process until all the fish filets are breaded. Tip: if you're making the fish ahead, at this stage you can cover and refrigerate the fish up to 24 before frying.
To fry off the fish, add ½ cup of oil to a 12" skillet or enough oil to come ¼ inch up the sides of the pan you're using. Heat over medium heat. You’ll know the oil is hot enough when you drop a few breadcrumb into the oil and it sizzles right away. At the same time, preheat the oven to 200℉.
Working in batches, fry the fish filets. Do not overcrowd the pan. Fry each piece of fish for 2 to 4 minutes on one side, flip and continue cooking the other side for another 2 to 4 minutes. The fish is cooked when the breading is golden brown and the internal temperature reaches 145°F on a meat thermometer. Adjust the temperature if the fish is browning too quickly.
Transfer the cooked fish to a sheet pan in a 200℉ oven to keep warm while you fry off the rest of the cod filets.
After the fish is cooked, toast the hamburger buns until golden around the edges. Add lettuce to the bottom bun and top with a piece of fried fish. If desired, add a slice of cheddar cheese to the hot fish, and spoon over the tartar sauce. Top with the second half of the bun and serve hot.