Stir all ingredients together in a small bowl. Taste-test and add more hot sauce if needed.
Make the vegan fried green tomatoes:
In a shallow dish, whisk together the milk, flaxseed, and flour. Set aside for 5-10 minutes to thicken up. Meanwhile, slice the tomatoes about ½-inch thick. Arrange the slices on a plate and sprinkle with salt. Stir the cornmeal and paprika together in a shallow dish.
Generously coat the bottom of a large skillet with oil and place it over medium heat. When the oil is hot, grab a tomato slice and dip both sides into the flaxseed batter. Transfer the slice to the dish with the cornmeal and dredge both sides, then transfer the slice to the skillet. Repeat with as many additional slices as you can fit into the skillet without crowding. Cook the slices until browned and crisp, about 5 minutes on each side, then transfer them to a paper towel-lined plate. Repeat until all of the tomato slices are cooked, adding oil to the skillet as needed between batches.
Make the Cajun red bean patties:
Place all ingredients except the oil into the bowl of a food processor and pulse until well chopped and incorporated, stopping to scrape down the sides of the bowl as needed. Taste-test and adjust the amount of hot sauce and salt if needed. Shape the mixture into 4 patties.
Cook the patties in the same skillet as the tomatoes, adding oil if needed so the bottom is lightly coated. Place the skillet over medium heat and add the patties in an even layer. Cook about 4 minutes on each side, until lightly browned. Work in batches if needed and add oil to the skillet between batches if it dries up.
Slather the buns with vegan remoulade. Stuff with patties, fried green tomatoes, and top with lettuce, radicchio, and red onion. Serve.