Add the chopped onion to a food processor fitted with a metal blade. Process until the onion is pulverized and almost liquified. Add the cod to the food processor with the onion and pulse until the fish is broken down into a puree with some small pieces in it. No pieces should be larger than pea-sized. Scrape the contents into a mixing bowl and stir in the bread crumbs, chives, parsley, eggs, and Old Bay until it is evenly combined and homogenous.
Place a 12-inch nonstick skillet over medium-high heat. Lavishly butter the cut sides of all of the sandwich rolls. Toast them, as many as you can comfortably fit in a pan, until deep golden brown. Transfer the rolls to a plate and repeat with the remaining roll halves.
Add 1/8-inch of oil to the pan. Swirl the oil to coat the pan. Allow the oil to heat until it is shimmery and hot (this can be tested by dropping a small torn piece of bread into the oil. If the oil bubbles around the bread immediately, it is ready.)
Don’t pre-form all your fish cakes, because the patties tend to fall apart if left to rest on a plate It’s best just to form right before adding them to the pan. To do this most easily, divide the fish mixture into 8 evenly sized amounts. Only form as many of those into patties as you can fit into the pan without crowding. Don’t squash them mercilessly! A light end texture relies on a gentle hand.
Once your first batch of patties are in the pan, raise the heat to high and cook for 4 to 5 minutes, or until the bottoms are a deep brown. Slide a wide spatula under the fish burgers and lay an additional rubber spatula on top of the burgers. Use both spatulas to flip the patties one at a time. Take care to control how your flip them so you’re not splattered with any hot oil in the pan. Transfer that batch of burgers onto a plate, lightly tent with foil to retain heat, and repeat with the remaining fish burger mixture. Transfer each burger to a toasted bun. Top with coleslaw and serve immediately.