To make the sauce:
In a small bowl, combine the mayonnaise, ketchup, sweet pickle relish, smoked paprika, garlic powder, salt, and black pepper. Stir thoroughly until the mixture looks smooth and evenly blended, with no streaks of mayonnaise remaining. Taste the sauce and adjust the seasoning if needed, then cover the bowl and place it in the refrigerator until serving time. Keeping the sauce chilled helps the flavors settle and gives the final dip a fresher taste.

To assemble the quesadillas:
Lay 6 tortillas out on a clean work surface. On each tortilla, arrange 1 slice of Swiss cheese so it covers as much of the surface as possible. Add an even portion of corned beef on top of the cheese, spreading it out so the filling stays balanced from edge to edge. Spoon the drained sauerkraut over the meat, distributing it evenly across the surface. Place the remaining Swiss cheese slices over the sauerkraut, then top each one with a second tortilla to form a sandwich-style stack.

To cook the quesadillas:
Set a large skillet over medium-high heat. Once the pan is hot, add about ½ tablespoon of oil and swirl it around so the bottom is lightly coated. Carefully place one quesadilla in the skillet. Cook it for several minutes on the first side, until the tortilla turns golden brown and crisp and the cheese underneath begins to melt. Use a spatula to flip the quesadilla gently, then cook the second side until it is also browned and the cheese has fully melted through the center.

Transfer the cooked quesadilla to a rack or a warm plate while you repeat the process with the remaining quesadillas, adding a little more oil to the skillet as needed. Keep the heat steady so the tortillas brown evenly without scorching. Once all of the quesadillas are cooked, let them rest for a moment, then slice each one into wedges. Serve immediately with the chilled 1000 Island sauce on the side for dipping.