In a large bowl, combine the chicken thighs and oil. Toss to coat the thighs evenly.
Transfer the thighs to a platter and sprinkle evenly on all sides with the cumin rub. Set aside at room temperature.
Prepare a charcoal or gas grill for indirect grilling over medium heat; the temperature inside the grill should be 350°—375°F (180°—190°C). If using charcoal, bank the lit coals on either side of the grill bed, leaving a strip in the center without heat, and place a drip pan in the center. If using gas, preheat the burners, then turn off 1 or more of the burners to create a cooler zone.
Brush and oil the grill grate.
To make the salsa, place the tomatillos and chile over the direct-heat area of the grill. (You'll probably need a grill screen for the chile or you'll lose it through the grate.) Cook, turning as needed, until well charred on all sides, about 3 minutes.
Transfer the tomatillos and chile to a blender, add the lime juice and oil, and pulse until combined but still chunky.
Transfer to a bowl, fold in the cilantro, and season with salt. You should have about 2 cups (16 fl oz/500 ml). (The salsa can be made up to 1 week in advance and refrigerated. Bring to room temperature before serving.)
Place the thighs, meaty side down, over the indirect-heat area of the grill. Cook, turning once, until the thighs are nicely grill-marked on both sides and firm to the touch and an instant-read thermometer inserted into the thickest part of the thigh away from bone registers 170°F (77°C), 10—15 minutes on each side.
Transfer the thighs to a platter and let rest for 10 minutes.
Serve at once with the tomatillo salsa.
The Game Plan
Make rub; coat chicken with oil and season with rub Set up grill for indirect grilling over medium heat Make tomatillo salsa Grilling time: 20–30 minutes DON'T FORGET: Try to turn the thighs only once so they develop nice grill marks.
