Peel pineapple and cut into 1/2-inch slices. Use a small round cookie cutter to cut the core out of each piece.
In a large skillet, melt butter with cinnamon stick over medium heat. Cook until butter starts to brown, about 2 minutes.
Add pineapple slices and cook 3 to 4 minutes per side. Remove pan from heat and add rum and brown sugar. Place pan back on heat and cook over medium-high heat until juices are thick and syrupy. Transfer pineapple to a plate and reserve syrup.
Heat oven to 325 degrees and set a wire rack on a rimmed baking sheet.
To make batter, whisk flour, sugar, baking soda, and salt.
Add buttermilk, egg, melted butter, and vanilla. Whisk until blended.
Heat a large skillet or griddle over medium heat. Coat with cooking spray. For each pancake, pour 1/4 cup batter onto skillet. Let cook until bubbles form on top. Place pineapple slices on top and carefully flip pancakes over. Cook for 2 minutes and then place on prepared rack. Place rack in oven until cooked through, about 5 minutes.