Place first 5 ingredients in a mini food processor. Pulse 10 times or until coarsely chopped. Stir in 1/8 teaspoon salt; set aside.
Heat a small skillet over medium heat. Add oil; swirl to coat. Add onion to pan; sauté 3 minutes or until tender. Cool slightly. Combine onion, breadcrumbs, parsley, sage, pepper, turkey, and remaining 1/4 teaspoon salt in a large bowl, stirring just until combined. Divide mixture into 4 equal portions, shaping each into a 4-inch patty.
Heat a grill pan over medium-high heat; coat with cooking spray. Add patties to pan; grill 5 minutes. Turn patties; grill 3 minutes. Top each with 1 cheese slice; grill 2 minutes or until cheese melts and turkey is done.
Divide arugula evenly among bottom halves of buns. Top each serving with 1 patty. Divide relish evenly among servings; top with top halves of buns.
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