Start by trimming the drumsticks. Using sharp kitchen shears, carefully cut through the skin and connective tissue around the base of the bone. Push the meat downward toward the opposite end, creating a lollipop-like appearance with about 3 inches of bone exposed. The result should be a rounded, compact bundle of meat at one end of the bone.
Place the trimmed chicken lollipops into a large resealable plastic bag. Pour in one cup of garlic salad dressing, seal the bag, and gently shake to ensure all pieces are evenly coated. Refrigerate the bag for at least 6 to 8 hours, allowing the marinade to infuse the chicken with bold garlicky flavor.
When ready to cook, preheat the grill to medium heat—aim for a consistent temperature around 350°F.
Remove the lollipops from the marinade and discard the excess liquid. Season the chicken generously with beef rub or a mixture of salt and pepper. Place the lollipops directly onto the grill grates, close the lid, and cook for 30 to 35 minutes, turning the drumsticks every 10 minutes to achieve even browning on all sides.
While the chicken is grilling, prepare the glaze. In a small saucepan over medium heat, combine ½ cup garlic salad dressing, ½ cup of BBQ sauce, and ¼ cup of honey. Stir frequently as the mixture heats, allowing it to blend and thicken into a glossy, sticky glaze—this should take about 10 minutes. Once thickened slightly, remove the sauce from heat and let it cool for a few minutes.
Once the chicken is cooked through and nicely browned, transfer the drumsticks to a large bowl and pour the warm glaze over them. Toss well to coat each lollipop thoroughly with the sticky sauce.
Return the glazed drumsticks to the grill and reduce the heat to low. Continue cooking for another 10 minutes, or until the internal temperature of the meat reads 165°F. This final stage helps set the glaze and create a deliciously lacquered finish.
Transfer the lollipops to a serving platter. For an extra touch of sweetness, drizzle with additional honey before serving, if desired.