Combine 1/4 cup water with the brown sugar, soy sauce, fish sauce, lemongrass, garlic and shallot in a shallow dish. Add the pork chops and turn to coat. Let marinate 15 minutes.
Meanwhile, cook the rice as the label directs, adding the grated ginger to the water.
Preheat the broiler. Remove the pork chops from the marinade; transfer to a broiler pan (reserve the marinade). Broil until the pork is charred and cooked through, 6 minutes. Transfer the marinade to a skillet; add 1/4 cup water and bring to a boil over high heat. Cook, stirring occasionally, until thickened, about 3 minutes.
Fluff the rice with a fork and stir in the mint and cucumber; season with salt and pepper. Serve the pork with the rice and pan sauce.