Pre-heat a grill to medium heat.
Place ripe plantains, with skin on, on grill for 15 minutes, or until skin is dark brown. Remove from grill. Slice plantains and peel off skin. Coat each with glaze and return to grill. Grill for 2 more minutes, or until plantains are a golden brown.
Remove plantains from grill. Serve on a plate with cinnamon ice cream.
In a medium saucepan, melt butter over medium heat. Add brown sugar and cinnamon and stir until sugar dissolves. Add rum and cook for 2 minutes. Remove from heat.
Combine cream, milk, cinnamon sticks, and brown sugar in a large saucepan over medium heat. Scrape seeds from vanilla bean into the pan and then add whole bean. Bring to a boil, and then remove from heat and let steep for 5 minutes.
In a medium mixing bowl, whisk sugar and yolks together until pale yellow. Gradually whisk in about 1/3 of the hot cream mixture to temper the eggs. Return the mixture to the saucepan. Stir over medium-low heat until custard thickens, about 4 minutes (do not boil). Strain into a large bowl. Whisk in ground cinnamon. Chill until cold, about 3 hours.
Transfer cold custard to an ice cream maker and process according to the manufacturer's instructions.