Place Tbsp coarse sugar in a small bowl. Slice tangerines or clementines in half and dip cut sides into the sugar.
Place tangerines cut-side down in a cast-iron skillet over medium-high heat.Add about 6 sprigs of rosemary in pan. Let cook for about 2 minutes, until sugar on tangerines caramelizes. Remove from heat.
Discard rosemary (or use for another purpose). Let tangerines cool a bit and then cut each half into quarters.
Put half the tangerines and 1 tsp remaining sugar in a cocktail shaker. Muddle to release as much juice as possible from tangerines. Add 2 oz. rum and crushed ice to shaker.
Shake well and pour into ice-filled glass. Repeat with remaining ingredients. Garnish glasses with additional rosemary.