Toast the rice in an 8-inch skillet over medium-low heat, stirring frequently, until golden, 4 to 5 minutes (the rice will begin to smoke after a couple of minutes). Let the rice cool slightly and then grind in a spice grinder until the largest pieces resemble very coarse cornmeal; the mixture should not be completely powdery.
Combine the fish sauce, lemongrass, pepper flakes, brown sugar, and 1/2 cup water in a 12-inch nonstick skillet and bring to a simmer over high heat. Separate the chicken into large clumps and add to the pan. Cook, breaking up the chicken into small pieces with a wooden spoon, until the meat is no longer pink, 5 to 6 minutes. Sprinkle 1 Tbs. of the ground rice over the chicken and continue to cook, stirring frequently, until the liquid in the pan has thickened, about 2 minutes longer.
Remove from the heat. Stir in the shallot. Sprinkle with the lime juice, scallions, cilantro, and mint and stir gently to combine. Transfer to a serving dish and sprinkle with 1 tsp. of the remaining ground rice. Serve with the lettuce leaves on the side to use as cups for the chicken.