In a medium bowl, whisk together the mayonnaise, spicy brown mustard, paprika, parsley, horseradish, lemon juice, Cajun seasoning, pickle juice, hot sauce, grated garlic, and black pepper. Taste the sauce and adjust the seasoning if needed. Cover and refrigerate until ready to serve.
Using a serrated knife, slice the green tomatoes into 3/8-inch thick slices. Sprinkle both sides of the tomato slices with coarse salt, then let them stand for 30 minutes. This helps pull out extra moisture so the coating sticks better. Gently pat the tomatoes dry with paper towels before breading.
Set up three shallow dishes for breading. In the first dish, whisk together the flour, coarse salt, smoked paprika, garlic powder, onion powder, and black pepper. In the second dish, whisk together the buttermilk, egg white, and hot sauce. In the third dish, combine the panko breadcrumbs (or ritz crackers) and fine white cornmeal (or cornstarch).
Fit a wire rack inside a rimmed baking sheet. Working one slice at a time, dip each tomato slice into the flour mixture and shake off the excess. Dip it into the buttermilk mixture, then coat it in the cornmeal (or cornstarch) and panko mixture, pressing gently so the coating sticks.
Place the breaded tomato slices on the prepared wire rack in a single layer. Let them sit while the oil heats. This short rest helps the breading set, which makes the tomatoes easier to fry and helps keep the coating from falling off.
In a large cast iron skillet, heat the vegetable oil over medium-high heat until it reaches 350°F. Carefully add 3 to 4 tomato slices to the hot oil, making sure not to overcrowd the skillet. Fry for 3 to 5 minutes per side, or until the tomatoes are golden brown and crisp.
Transfer the fried green tomatoes to a paper towel-lined plate to drain. Continue frying the remaining tomato slices in batches, adjusting the heat as needed and replacing the oil if necessary.
Serve the fried green tomatoes immediately with the chilled remoulade sauce on the side.