Preheat the oven to 375°F and line a small rimmed baking sheet with aluminum foil.
Add the ground turkey, mayonnaise, horseradish, breadcrumbs, salt and a few grinds of black pepper to a mixing bowl. Combine gently but thoroughly and form the mixture into four 1/2-inch thick patties and chill for 15 minutes.
While the burgers chill, make the Russian Dressing by combining the mayonnaise, chili sauce, horseradish and onion in a small bowl. Set aside.
Coat the bottom of a large frying pan with a small amount of vegetable oil and heat over medium high heat. Working in two batches if necessary, sear the turkey burgers until lightly browned, 1 minute per side.
Transfer the burgers to the prepared baking sheet and bake for 4 minutes. Remove from the oven, top each burger with a slice of cheese and bake for an additional 1 to 2 minutes until the cheese has melted and the burgers are cooked through.
Remove the burgers from the oven and let rest for 5 minutes.
While the burgers rest, wipe out the pan, add the butter and melt over medium heat. Add the sauerkraut, combine well and cook for 2 minutes until heated through.
To serve, place a burger on each bun and top with sauerkraut and Russian dressing.