In a bowl, mix the fish sauce, brown sugar, minced garlic, shallots, oil, dark soy and black pepper. Split the marinade between the ground pork and the sliced pork shoulder in two separate bowls. Mix each well and let them sit for at least 30 minutes, or overnight in the fridge.
Make the nuoc cham. Stir the warm water and sugar until dissolved, then add the fish sauce, lime juice, garlic and chili. Taste and balance. It should be bright, salty and lightly sweet. Stir in the julienned carrot and set aside.
Shape the ground pork into small flat patties, about 2 tablespoons each. Keeping them thin helps them char fast without drying out.
Cook the rice vermicelli per the package, usually 4 to 6 minutes in boiling water. Drain, rinse under cold water until cool, and set aside so the strands stay separate.
Heat a grill pan or heavy skillet over high heat until it just starts to smoke. Grill the patties and the pork slices in batches, without crowding, until deeply charred on both sides, about 3 to 4 minutes per side. Do not move them too early. You want real color.
Drop the grilled pork straight into the bowls of nuoc cham so it soaks up the sauce while still hot. Serve with the vermicelli, lettuce, herbs and bean sprouts on the side. Each person dips noodles and herbs into the pork and broth bowl as they eat.