Bring 4 qt. water to a boil in a Dutch oven over high heat. Reduce heat to low. When water is just simmering, gently lower eggs into water using a skimmer or ladle. Simmer 14 minutes. Drain immediately, and return eggs to pan. Fill pan with cold water and ice. Let stand until cool enough to handle (about 10 minutes).
Meanwhile, place wood chips in center of a broiler pan. Wrap broiler rack with heavy-duty aluminum foil, and lightly coat top with cooking spray. Place rack in broiler pan.
Remove eggs from ice water. Tap each egg firmly on the counter until cracks form all over the shell. Peel under cold running water. Cut eggs in half lengthwise, and carefully remove yolks. Place yolks in a medium bowl. Place egg whites on rack of broiler pan.
Partially cover broiler pan tightly with heavy-duty foil, folding back 1 corner to create a 1/2-inch opening. Place over medium-high heat on stove-top. When you see the first wisp of smoke, cover tightly with foil. Smoke 6 minutes. Remove from heat; let stand 2 minutes. Remove whites from pan; coarsely chop.
Press yolks through a fine wire-mesh strainer into a bowl. Stir in mayonnaise and next 3 ingredients until mixture is smooth and resembles cake batter. Fold in egg whites.
Toast bread; cut in half diagonally. Top with egg salad. Sprinkle with freshly ground pepper to taste, and top with arugula.
Note: We tested with Hellman's Mayonnaise and Whole Foods Market Prairie bread. Purchase indoor smoking chips at kitchen emporium.com.
Adapted from The Lee Bros. Charleston Kitchen, copyright 2013 by Matt Lee and Ted Lee.