Peel bananas and slice diagonally, 1/4 inch thick. Pour orange juice into a small, shallow bowl. In another small bowl, mix brown sugar and cinnamon.
Melt 1 Tablespoon butter in a small frying pan over medium heat. Dip a third of the banana slices into the orange juice and then into the brown sugar mixture. Add to the pan and cook until lightly browned and glazed on both sides (about 1 minute total). Spoon into 2 shallow dessert dishes. Repeat two more times with the remaining bananas.
When all are cooked, add lime juice and any remaining orange juice and brown sugar mixture to the pan. Cook over medium heat, stirring, until mixture boils and becomes syrupy (this happens quickly). Pour evenly over bananas.
Top each serving with a dollop of sour cream. Serve immediately.