Make the Sauce

1. In a medium bowl, stir together 1/2 cup mayonnaise, 2 tablespoons ketchup, 2 tablespoons finely chopped kosher dill pickle, 1 tablespoon sauerkraut brine, 1/2 teaspoon Worcestershire sauce, 1/2 teaspoon Dijon mustard, 1 teaspoon prepared horseradish, 1/4 teaspoon onion powder and 1/4 teaspoon smoked paprika until fully combined.
2. Taste and adjust as needed: add a few extra drops of brine for more tang, a little more horseradish for heat, or additional mayonnaise if you prefer a thicker sauce.
3. Let the sauce sit for 10 to 15 minutes while preparing the quesadillas to allow the flavors to meld.

Make the Quesadillas

1. Warm a large skillet over medium heat. If using, add 1 1/2 teaspoons lightly crushed caraway seeds and toast, stirring constantly, until fragrant, about 30 seconds. Add 8 ounces thinly sliced corned beef, roughly chopped, and cook for 2 to 3 minutes until warmed through. Set aside to cool slightly. 
2. In a large bowl, add 6 ounces sauerkraut (drained and squeezed very dry). Taste; it should have a bright, acidic, salty bite. If it needs more tang, add 1 to 2 teaspoons sauerkraut brine, 1 teaspoon at a time, stirring and tasting after each addition, until the flavor is sharp. Season with salt only if needed.
3. Add 5 ounces shredded Gruyère cheese and 3 ounces shredded sharp white cheddar to the sauerkraut. Once the corned beef has cooled slightly, fold it into the sauerkraut mixture.
4. Lay out 4 (8-inch) flour tortillas. Divide the filling evenly over one half of each tortilla, pressing it down into an even layer and leaving about a 1/4-inch border along the edge. Fold the empty half over the filling and firmly press. Brush the tops lightly with melted butter.
5. Heat a large skillet over medium-low heat. Place the quesadillas butter-side down in the pan, one or two at a time depending on skillet size. Press gently with a spatula and cook until golden brown and crisp, about 2 to 3 minutes.
6. Brush the top with melted butter, flip, and cook 2 to 3 minutes more, until the second side is golden and the cheese is melted.Transfer to a cutting board and let rest for 1 to 2 minutes. Repeat with remaining quesadillas.
7. Slice and serve warm with the sauce.