In a small bowl, whisk the fish sauce with the brown sugar, lime juice, rice vinegar, chopped cilantro stems, chopped mint, pepper, garlic and, ¼ cup of water. Set the fish sauce vinaigrette aside.
In a large pot of salted boiling water, cook the rice noodles according to the package instructions. Drain and rinse under cold water. Transfer the noodles to a rimmed baking sheet and drizzle with the oil and 2 tablespoons of the fish sauce vinaigrette. Toss to coat.
When you’re ready to cook, preheat a grill to medium (400°F). Clean the grate thoroughly and brush with oil. Alternatively, set a grill pan over medium-high heat and brush with oil. Grill the skewers over direct heat with the lid closed, turning once, until well browned and cooked through, 10 to 12 minutes. Transfer to a plate to rest.
Divide the rice noodles, romaine, scallions, cucumber, and carrots among shallow bowls. Top each bowl with a skewer. Drizzle with some of the fish sauce vinaigrette and garnish with small cilantro sprigs, torn mint leaves, crispy shallots, and chopped nuts. Alternatively, arrange the rice noodles, vegetables, skewers, and garnishes on a large platter and let people build their own bowls. Serve right away, passing lime wedges and more fish sauce vinaigrette at the table.