In a small bowl, combine all of the dressing ingredients.
Heat a griddle pan over medium-high heat. Combine all of the patty ingredients except for the oil; form 4 patties. Drizzle the oil on the griddle, add the patties and cook, turning once, until cooked through, 8-10 minutes. Transfer to a plate. Wipe the griddle.
In a small saucepan, warm the sauerkraut over medium heat.
Lightly spread 1 side of each bread slice with the butter. With the buttered sides facing down, build the patty melts: Spread the dressing liberally on unbuttered side of 4 bread slices, then top each with two slices of Emmenthaler slices, an even layer of sauerkraut, a turkey patty and 2 cheddar slices. Slather the mustard on the unbuttered sides of the 4 other slices and set in place. Griddle, turning once, until crisp on the outside and cheese is melted, 6-8 minutes.