Preheat a grill or large cast-iron grill pan to medium-high heat and oil the grate.
In a small bowl, whisk the lime zest and juice with 1/4 cup of the brown sugar, the soy sauce, garlic and 1/2 teaspoon each of salt and white pepper.
Spoon half the soy mixture into a 9-by-13-inch baking dish. Arrange the short ribs in an even layer on top, then spoon the remaining soy mixture on top; let marinate for 5 minutes.
Meanwhile, in a medium skillet, heat the peanut oil over medium-high heat. Add the peanuts, red chiles and the remaining 1 teaspoon brown sugar to the skillet. Cook, stirring constantly, until the peanuts and chiles are fragrant, 2 to 3 minutes. Spread on a small baking sheet or large plate to cool, stirring occasionally.
Grill the short ribs until lightly charred on the bottom, 3 to 4 minutes. Flip the short ribs and cook until almost cooked through, 3 to 4 more minutes per side; transfer to a platter.
Using scissors, cut the short ribs off the bones and into bite-size strips. Serve in lettuce leaves with the spicy peanuts, sliced scallions, shredded carrot, mint leaves, Thai or sweet basil leaves and lime wedges.