2 to 3 tablespoons vegetable oil plus more for grates
1/3 cup coarse salt
1/4 cup packed light-brown sugar
1/4 cup paprika
2 tablespoons ground black pepper
2 tablespoons dried oregano
2 tablespoons dried thyme leaves
1 tablespoon cayenne pepper (optional)
Spice Rub: In a small bowl, combine all the ingredients, using your hands to break up the sugar. Store in an airtight container, away from heat and light, up to 6 months.
In a shallow baking dish, place chicken drumsticks and thighs. Carefully slide fingers under skin to loosen it from the meat. Rub chicken all over with oil. Season with All-Purpose Spice Rub both under and over skin, patting in gently. Proceed to step 2 right away, or cover chicken and let rest for up to 24 hours. Cover and refrigerate if resting longer than 30 minutes.
Heat grill to medium-low. Oil grates; grill chicken, covered, turning frequently to prevent charring, until an instant-read thermometer inserted in thickest part (avoiding bone) registers 175 degrees, 20 to 30 minutes.