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Preheat oven to 425°. Whack cinnamon stick with the back side of a chef’s knife to break into small pieces. Toast cinnamon, allspice, peppercorns, and cloves in a dry small skillet over medium-high, tossing, until fragrant, about 2 minutes; let cool.
Grind spices, brown sugar, thyme, and cayenne in spice mill or with mortar and pestle to a fine powder. Mix oil and 2 Tbsp. spice rub in a small bowl (save remaining rub for another use). Rub chicken legs with jerk spice mixture; season with salt. Place on a rimmed baking sheet and bake until golden brown and cooked through, 30–35 minutes.
Do Ahead: Jerk spice can be made 3 months ahead. Store airtight at room temperature.