In a small bowl, combine the lime juice, 2 tbsp. fish sauce and the brown sugar.
Preheat a grill to medium-high. In a large bowl, mix together the turkey, shallot, chile, lime zest and remaining 2 tsp. fish sauce. Form the mixture into eight 1/2-inch-thick patties.
Divide the lettuce among 4 plates. Top with the carrots and bell pepper. Brush the turkey patties on both sides with oil and grill, turning once, until browned and cooked through, about 6 minutes. Arrange 2 patties on top of each salad. Drizzle with the dressing and top with the mint.