Bring rinsed cup of rice and 1 1/4 cups water to a boil in a saucepan. Cover; cook on low 15 minutes, then rest covered 5 minutes.
Toast 2 tablespoons dry rice in a small skillet over medium heat 4 to 5 minutes, shaking often, until golden. Crush finely with mortar and pestle or spice grinder.
Put beef and 1/4 cup water in a large skillet. Cook over medium-high heat 7 to 9 minutes, stirring into fine crumbles, until no liquid pools and meat reaches 160°F.
Remove from heat. Mix lime juice, fish sauce, sugar, and red pepper; stir into beef with shallot and toasted rice powder.
Cool beef 2 minutes; fold in mint and cilantro. Spoon into lettuce leaves; serve cooked rice alongside, dividing into four portions.