Quick-pickle: whisk sugar, vinegar, water, and salt until dissolved. Add julienned carrots and daikon, press into the liquid, and refrigerate at least 1 hour (or overnight).
Marinate the pork: stir together fish sauce, sugar, soy sauce, 2 tablespoons oil, lemongrass, garlic, shallot, and black pepper. Coat the chops, cover, and refrigerate at least 2 hours or overnight. Bring to room temperature 20-30 minutes before grilling.
Cook the broken rice: rinse in several changes of water, combine with 1 3/4 cups water and salt, and cook like jasmine rice. Rest covered 10 minutes, then fluff.
Make the nuoc cham: dissolve sugar in warm water, then add fish sauce, lime juice, garlic, and chili. Adjust to taste.
Steam the egg meatloaf: mix ground pork, 2 whole eggs plus 2 whites, fish sauce, sugar, pepper, and soaked mushrooms/noodles if using. Press into an oiled heatproof dish and steam 20 minutes. Brush the 2 reserved yolks over the top and steam 5 minutes more until set and golden. Cool slightly, then slice.
Make the bi: toast the sticky rice in a dry skillet until golden and nutty (5-7 minutes), cool, and grind coarse. Simmer the pork strips in salted water 12-15 minutes until cooked through, cool, shred thin, and toss with the rice powder, salt, and pepper.
Grill the pork: heat a grill, grill pan, or broiler to high and oil the grates. Grill chops 4-5 minutes per side until cooked through with charred, caramelized edges. Rest 5 minutes, then slice into strips.
Make the scallion oil: pour the hot shimmering oil over the sliced scallions in a heatproof bowl and stir once.
Assemble: mound broken rice on each plate, fan out sliced pork, and add cha trung, bi, pickles, cucumber, tomato, and lettuce. Spoon scallion oil over the rice and serve nuoc cham on the side.