In a medium bowl, toss the carrots, daikon, lime juice, sugar, and 1/2 teaspoon salt. Let stand while you make the burgers.
In a large bowl, combine the pork, scallions, lime zest, garlic, ginger, and 1/2 teaspoon salt with your hands until well mixed. Form into four 1/2-inch-thick patties that are slightly larger than your buns.
Heat a large cast-iron skillet over medium heat. Rub it with oil to coat.
If it's large enough for all 4 patties, without overcrowding, then cook all 4 at once. Depending on the size of your pan, you may have to cook 2 at a time.
Cook until well browned, about 6 minutes, then flip and cook until the meat is cooked through, about 6 minutes longer.
Repeat with the remaining patties.