In small bowl, mix turkey, cayenne, paprika, and salt. Form into 6 patties Press thumb in center of patties so the burger will not bulge in the middle. Place on grill and grill each side approximately 6 minutes or until no pink remains in the center.
For the avocado relish, lightly mash avocado with onion, jalapeño, lime juice, and cilantro. The avocado mixture should be creamy but slightly lumpy.
To assemble, place burger on bun. Top with 1/2 cheese slice, 1 tablespoon avocado relish, and arugula. Serve.