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