To prepare relish, combine first 8 ingredients in a medium bowl. Cover and chill 2 hours.
To prepare burgers, place beans in a large bowl; partially mash with a fork. Stir in breadcrumbs and next 5 ingredients (breadcrumbs through egg). Divide bean mixture into 4 equal portions, shaping each into a 1/2-inch-thick patty.
Place patties on grill rack coated with cooking spray; grill 5 minutes on each side or until thoroughly heated. Spread 1 tablespoon mayonnaise on the bottom half of each bun; top each with a patty, 1/4 cup relish, and top half of bun. Garnish with dill sprigs, if desired.