Step 1Peel and finely-julienne the carrot and daikon, and place in a glass or non-reactive container or bowl.
Step 2In a saucepan, heat the sugar, water, vinegar and salt over high heat, stirring until the sugar and salt are dissolved. Continue heating just until the mixture comes to a boil. Remove from heat and pour over the vegetables.
Step 3Stir the mixture, coating the vegetables and making sure they are submerged (if necessary, weight them down to keep them under the liquid). Set aside for ½ to 1 hour to quickly pickle, then cover and refrigerate. This makes about 3 cups drained pickles, which will keep, refrigerated, up to 2 weeks.
Step 1Heat a griddle or cast iron pan over medium-high heat until hot. Add the shishito peppers and toast until softened and blistered on all sides, 4 to 6 minutes, rotating frequently. Remove from heat and set aside until cool enough to handle.
Step 2Butter 1 side of each of the slices of bread with mayonnaise. Top each slice with half of the cheese. On top of one slice, arrange the shishito peppers, then the drained pickles and cilantro sprigs. Top with the inverted second slice of bread to form a sandwich, pressing gently so everything stays together.
Step 3Heat the griddle over medium-high heat until hot. Meanwhile, butter the outside of one side of the sandwich with mayonnaise.
Step 4Invert the sandwich onto the hot griddle, and butter the second (top) side with mayonnaise. Grill the sandwich until the bread is crisp and golden-brown, and the cheese is starting to melt. Flip the sandwich and repeat with the second side. Remove from heat.
Step 5Slice the sandwich and serve immediately.