Heat the oil in a large skillet over medium-high heat. Add the pork and cook, breaking it up with a spoon, until no longer pink, 2 to 3 minutes.
Add ¼ cup water, 1 tablespoon of the fish sauce, and 1 teaspoon of the sugar and cook, stirring, until the liquid is almost evaporated, 2 to 4 minutes more.
Stir together the lime juice, jalapeño peppers, and the remaining 4 tablespoons of fish sauce and 2 teaspoons of sugar in a small bowl.
Divide the lettuce among bowls, top with the pork mixture, shallot, carrots, mint, and peanuts, and drizzle with the lime dressing.