Preheat the broiler. Toss the carrots, daikon, jalapeño, vinegar, sugar, 1 teaspoon salt and a few grinds of pepper in a medium bowl; set aside.
Cook the bacon in a large nonstick skillet over medium-high heat, turning, until crisp, 6 to 8 minutes. Remove to a plate. Arrange the rolls cut-side up on a baking sheet. Drizzle the bottom halves with some of the bacon drippings; reserve the skillet. Broil the rolls until golden, 1 to 2 minutes. Top with the cucumber slices. Halve the bacon slices and divide among the rolls.
Heat the reserved skillet with any remaining drippings over medium-high heat. Working in batches, crack in the eggs and cook until the whites are crisp around the edges and the yolks are still runny, about 5 minutes. Season with salt and pepper. Divide among the roll halves.
Add the cilantro to the pickled vegetable mixture and toss. Top each sandwich with some of the pickled vegetables and their juices.
Photograph by Ryan Dausch