Combine ground turkey, garlic, thousand island dressing, and 1/2 cup Italian-style bread crumbs in a bowl. Do not overwork the mixture. Form into 4 patties; transfer to plate. Sprinkle with garlic salt and additional 1/4 cup bread crumbs; press crumbs into patties to adhere.
Heat olive oil in a large non-stick skillet over medium-high heat. Cook patties in oil until golden brown, 7 to 10 minutes per side. Transfer to plate; top each patty with a slice of mozzarella cheese.
Cook and stir onion rings in same skillet until tender, about 10 minutes, adding oil if necessary. Sprinkle with ground black pepper.
Heat a non-stick griddle to medium heat. Lightly butter bread slices on both sides; cook on each side until browned, about 2 minutes per side.
To assemble burgers, layer turkey and cheese patties, sauerkraut, and cooked onions between bread slices.
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