Prepare the fish: Preheat the oven to 375 degrees. Drizzle a sheet pan with some of the olive oil and arrange the fish fillets on the pan. Drizzle the remaining olive oil over the fish and season with salt and pepper. Place the pan in the oven and cook for about 5 minutes, or until the fish is barely cooked through and just beginning to flake. Let cool, then using your fingers, gently break the fish into big bite-size chunks. Refrigerate until ready to use.
Whisk the eggs and mayonnaise together in a bowl until smoothly combined. Add ½ cup of the bread crumbs along with the corn, country ham, scallions, and largemouth pieces. Season with salt and pepper. Using a rubber spatula, gently fold the ingredients together, trying not to break up the fish any further, until just combined. Gently, with wet hands, form the mixture into eight patties and transfer to a sheet pan lined with parchment or wax paper. Refrigerate the patties for at least 30 minutes to firm up.
While the patties are chilling, make the remoulade: Grate the cucumber using the large holes of a box grater. Transfer to a strainer, toss with about ½ tsp. salt, and allow to sit for about 20 minutes. In a small bowl, combine the cucumbers with the yogurt, mustard, capers, pickles, lemon zest, and Tabasco. Salt and pepper to taste.
Put the remaining bread crumbs on a plate and season generously with salt and pepper. Heat the vegetable oil in a large sauté pan or skillet over medium heat; make sure there's enough oil to fully coat the bottom of the pan. Gently press the patties into the bread crumbs, on both sides, then transfer to the skillet. Cook for 4 minutes per side, lowering the heat if the bread crumbs brown too fast. Remove the patties when they are golden brown. (If working in batches, you may need to add more oil to the second batch.)
To serve, divide the remoulade between four plates and top with the cakes. Garnish with the chives and serve.