In a medium skillet over medium-high heat, brown the ground beef until fully cooked through, approximately 7-9 minutes. Drain excess fat.
Season the cooked beef with salt and black pepper; set aside.
In a small mixing bowl, whisk together mayonnaise, mustard, ketchup, and sweet relish to create the burger sauce.
Lay one large flour tortilla on a flat work surface. Spread 1 tablespoon of the burger sauce in the center (about the diameter of the tostada shell).
Place 1/4 of the seasoned ground beef over the sauce, then sprinkle with 1/4 cup of shredded cheddar cheese.
Place one corn tostada shell directly on top of the cheese layer.
Layer 1/4 cup of shredded lettuce, 2 tablespoons of diced tomatoes, and several pickle slices on top of the tostada.
Place a small flour tortilla on top of the vegetable layer.
Fold the edges of the large flour tortilla toward the center, overlapping the small tortilla to create a hexagonal shape and seal the wrap.
Heat vegetable oil in a large non-stick skillet over medium heat.
Place the wrap seam-side down in the skillet. Cook for 2-3 minutes until the bottom is golden brown and sealed.
Flip the wrap and cook the other side for an additional 2 minutes. Repeat the process for the remaining three wraps.