Put the carrots in a medium bowl and sprinkle with 1-1/2 tsp. of the sugar and 1/2 tsp. of salt. Using your hands, toss the carrots in the salt and sugar to begin expelling the water from them, about 3 minutes. They will soften and liquid will begin to pool in the bottom of the bowl. The carrots are ready when they can bend easily without snapping in half and they have lost about a third of their volume. Drain in a colander and rinse under cold water. Gently press down on the carrots to push out any additional water.
In the same bowl, combine 2 Tbs. of the sugar, the vinegar, and 1/4 cup of lukewarm water and stir to dissolve the sugar. Add the carrots to the pickling liquid (the liquid should be covering the carrots) and let sit for about 20 minutes.
Meanwhile, combine the pork, chili garlic sauce, fish sauce, lime juice, and the remaining 1-1/2 tsp. sugar. Shape into four 4-inch wide, 1/2-inch-thick patties and let sit at room temperature for 15 minutes.
Prepare a gas or charcoal grill or a grill pan at medium-high heat. Brush and oil the grates.
Make a thumbprint in the center of each of the burgers and then place on the grill. Grill the first side until grill marks form, about 4 minutes. Flip and continue to cook until an instant-read thermometer reads 145°F, about 3 minutes more.
Remove the burgers from the grill and let rest on a plate. Place the buns on the grill and toast until grill marks form on both sides, about 1 minute per side.
Spread a small amount of mayonnaise on the upper and lower halves of the buns. Lay a burger on the lower bun and top with a quarter of the carrot pickle, 4 cucumber slices, 4 jalapeño slices, a few cilantro leaves, a slice of ham, and the top half of the bun. Repeat with the other 3 burgers and serve.