As with wings, it’s important to remember that drumsticks require frequent turning (forget the notion of only flipping once) to cook evenly and create that beautifully crisp, browned skin that we all want to eat. Twenty drumsticks might seem like a lot, but trust me, they disappear quickly, and leftovers make for a very happy lunch (or picnic) the following day. —Paula Disbrowe
An hour before you plan to grill, soak the wood chips in water or beer. Place the drumsticks in a large bowl, drizzle with enough olive oil to lightly coat, season generously with salt and pepper, and sprinkle with the adobo seasoning, paprika, granulated garlic, and crushed red pepper and toss to combine. Set aside to marinate at room temperature while you prepare the grill. (Alternatively this can be done a day in advance and refrigerated in a large sealable plastic bag. Allow the drumsticks to come to room temperature before grilling.)
Prepare a grill for 2-zone cooking and build a medium fire in a charcoal grill, or heat a gas grill to medium-high. Carefully wipe the preheated grill grates with a lightly oiled paper towel. Using a grill brush, scrape the grill grates clean, then carefully wipe with a lightly oiled towel again.
Grill drumsticks over direct heat, turning and rotating around the heat as needed until all sides are nicely charred, 12 to 15 minutes, then transfer to a baking sheet.
Drain the wood chips and scatter them around the periphery of the hot coals (alternatively you can wrap them in a double layer of foil and poke holes in the top of the packet; remove the grill grate, place packet directly on coals, and replace the grate). If you’re using a gas grill, place the packet of chips (or use a smoker box) directly over the heat. When the chips begin to smoke, return the chicken to the grill over indirect heat. Cover grill and smoke chicken, turning every 5 minutes or so, until an instant-read thermometer inserted into the thickest part of the drumsticks registers 165ºF, 35 to 40 minutes more. Transfer the chicken to a serving platter, cover with foil to keep warm, and rest for 10 minutes before serving.