How to Make It
Whisk together first 3 ingredients in a large bowl. Whisk together egg and 1 1/2 cups water; add to cornmeal mixture, whisking until smooth.
Smash garlic to make a paste. Stir okra, jalapeño, and garlic paste into cornmeal mixture. (Batter will be thick and will thicken even more as it sits, so add water, if needed.)
Heat 1 Tbsp. oil in a large cast-iron skillet over medium heat. Pour 1 Tbsp. batter for each cake into skillet, and gently flatten into a 2-inch cake. (Don't overcrowd the cakes in the skillet.) Cook 2 to 3 minutes or until tops are covered with bubbles. Turn and cook 2 to 3 more minutes. Transfer to a paper towel-lined plate. Season with kosher salt and pepper. Keep warm in a 200° oven. Repeat procedure with remaining batter and oil.
Spread each cake with about 1 tsp. pimiento cheese. Top with arugula, tomato, basil, and kosher salt and pepper.