If the burgers brown too quickly, cover them with a heatproof metal bowl. This will help them to cook more quickly on the inside. Use tongs or a heavy potholder to remove the hot bowl from the grill.
To make the caramelized onions, heat butter and 1 tablespoon oil in a large skillet over medium-low heat. Add onions, cover, and cook until they begin to soften, stirring occasionally, about 10 minutes. Add sugar, 1/2 teaspoon salt, and 1/4 teaspoon pepper, and raise heat slightly. Cook, uncovered, until onions are golden brown, stirring occasionally, 25 to 30 minutes. Stir in thyme, and set aside.
Meanwhile, combine turkey, shallot, scallions, Tabasco, Worcestershire, remaining 1 1/2 teaspoons salt, and 1/2 teaspoon pepper in a large bowl; mix very lightly to combine.
Heat a grill or grill pan to medium high. Form four equal patties, and cook them through, about 7 minutes per side on the grill, or 8 minutes on the grill pan.
To toast the buns, brush the cut sides lightly with olive oil, and toast, cut side down, on the grill, until lightly golden, 20 to 30 seconds. Top burgers with the caramelized onions, and garnish with slices of tomato and lettuce leaves. Serve.