Make the apple mustard

1. Preheat a gas grill to medium (350°F to 400°F) on one side, or push hot coals to one side of a charcoal grill. Stir together apple, onion, olive oil, salt, and black pepper in a medium bowl. Transfer mixture to a large sheet of aluminum foil; fold foil over apple mixture, and seal edges to create a packet. Place foil packet on grates over unlit side of grill. Grill, covered, until apples are very soft, about 30 minutes.

2. Transfer apple mixture to a blender, and add 2 teaspoons mustard. Secure lid on blender, and remove center piece to allow steam to escape. Place a clean towel over opening; process until mixture is completely smooth, about 1 minute. Transfer mixture to a small bowl, and stir in remaining 2 teaspoons mustard. Cover, and chill until ready to serve.

Make the hot dogs

1. Increase heat or add additional charcoal to lit side of grill to bring to medium-high (400°F to 450°F). Poke a few holes in each sausage using the tip of a paring knife. Place sausages on grill grates over lit side of grill. Cook, uncovered, turning occasionally, until slightly charred, 3 to 4 minutes. Transfer sausages to unlit side of grill; cook, covered, until a thermometer inserted into center of sausages registers 160°F, 9 to 12 minutes.

2. To assemble hot dogs, place 1 sausage in each hot dog bun. Top with apple mustard and giardiniera; serve hot. (Reserve any remaining apple mustard for another use.)