Make the special sauce: in a small bowl, combine 1/3 cup mayonnaise, 1/3 cup ketchup, 2 tablespoons pickle juice, 1 tablespoon yellow mustard, 1 teaspoon Worcestershire sauce, and salt and freshly ground black pepper to taste (about 1/2 teaspoon salt and 1/4 teaspoon pepper). Set aside.
Cook the beef: in a large skillet over medium-high heat, cook 1 pound ground beef until browned and cooked through, about 6 to 8 minutes. Drain well.
Wipe out the skillet, coat with nonstick spray, and heat over medium. Working with 1 tortilla at a time, add a tortilla to the skillet. On half of the tortilla add 1/2 cup shredded cheddar cheese and 1/4 of the cooked beef, and top with dill pickle chips if using.
Sprinkle the top half with 1/2 cup shredded cheddar cheese, fold the tortilla in half, and cook until the cheese is melted and the bottom is golden brown, about 2 minutes.
Flip the quesadilla and continue cooking until the second side is golden brown and the cheese is fully melted. Transfer to a plate and cover to keep warm. Repeat with the remaining tortillas, cheese, beef, and pickles. Cut each quesadilla in half and stack 2 halves on each of 4 plates.
Assemble the wedge salads: add one cored wedge of Iceberg lettuce to each plate and top with the finely chopped shallot and quartered cherry tomatoes. Drizzle the quesadillas and salads with the special sauce and serve extra sauce on the side if desired.