Pickled Carrots and Daikon: Combine carrots, daikon, salt, and sugar in a small bowl and toss to coat. Allow to rest for 10 minutes until the liquid has been drawn out of the vegetables and the salt and sugar are completely dissolved. Add vinegar and water and transfer to a small, sealable container. Let sit at room temperature for 1 hour, or for best results, allow to rest for at least 3 days in the refrigerator.
Preheat oven or toaster oven to 400°F. Cook bacon in a 10-inch nonstick skillet over medium-low heat until crisp and rendered, flipping once halfway through cooking, about 10 minutes total. Transfer to a paper towel-lined plate (leave bacon grease in the skillet).
Increase heat to medium-high and heat until bacon fat starts lightly smoking. Add eggs, immediately reduce heat to medium, and cook, spooning hot bacon fat over egg tops as they cook until set to desired consistency (I like my yolks runny and white edges crisp). Transfer eggs to the paper towel-lined plate with bacon.
Split in half lengthwise, leaving a hinge. Toast bread in the oven or toaster oven until golden brown, about 3 minutes. Spread both sides with mayonnaise. Layer bacon strips along the bottom and top with eggs. Top with pickled carrots, daikon, cucumber slice, and cilantro. Season with salt and pepper. Serve immediately with clementine and apple juice.