Mix ground turkey, 1/2 cup salsa, shallots, cilantro, 3 tablespoons oil, hot sauce, cumin, salt, and pepper in large bowl. Shape turkey mixture into four 3 1/2- to 4-inch-diameter patties.
Heat 1 tablespoon oil in large nonstick skillet over medium-high heat.
Add burgers; cook until brown, about 3 minutes per side. Reduce heat to low. Sauté until burgers are cooked through, about 4 minutes, turning occasionally.
Arrange roll bottoms on 4 plates.
Place lettuce, then burgers on roll bottoms. Top each burger with 2 tablespoons of remaining salsa, then roll tops.
