Step 1: Set up your grill for direct grilling and preheat to high. Ideally, you’ll grill over a wood fire.
Step 2: Brush and oil the grill grate. Place the granulated sugar in a large mixing bowl. Dip the cut sides of the citrus fruits in the sugar, then grill cut sides down until the sugar caramelizes and the fruit is darkly browned, 3 to 4 minutes. Grill the cinnamon sticks until lightly toasted. Transfer the fruit and cinnamon to a cutting board and let cool.
Step 3: Dice the fruit into 1-inch pieces. Place in the bowl with the granulated sugar. Stir in the cinnamon sticks and brown sugar and muddle with a pestle or wooden spoon. Stir in the rum and let macerate for 10 minutes.
Step 4: Stir the prosecco into the fruit mixture. Correct the seasoning, adding sugar to taste. Ladle the sangria into ice-filled glasses and garnish with sprigs of fresh mint.