Step 1: Using poultry shears, cut the chickens in half. Cut out the backbones and cut the birds through the breastbones. Rinse under cold water and blot dry with paper towels. Generously season both sides with salt and pepper.
Step 2: Make the Horseradish Dip: Place the ingredients in a mixing bowl and whisk until smooth. Transfer 1 cup of the dip to an attractive bowl for serving. The combination may sound bizarre, but you’ll be amazed how good it actually tastes.
Step 3: Set up your grill for indirect grilling. If working on a gas grill, place the wood chips in the smoker box or in a smoker pouch and run the grill on high until you see smoke, then reduce the heat to medium-low (300 degrees). If working on a charcoal grill, toss half the wood chips on the coals.
Step 4: Arrange the chicken halves, bone side down, on the grill grate, over the drip pan (if using), away from the heat. Indirect grill for 45 minutes. Brush the chicken halves on all sides with some of the melted butter. Add charcoal or wood as needed. Continue indirect grilling until the chicken skin is brown and the meat is cooked and tender, another 30 to 45 minutes. The total cooking time will be 1-1/2 hours. Baste the chicken with melted butter one final time at the end.
Step 5: Place half the dip in a deep platter. Arrange the chicken halves on top and let steep in the dip for 3 minutes. Serve the reserved dip on the side.