Peel bananas and break them up or slice them. Place banana chunks into blender or food processor.
Add lemon juice and sugar and pulse until a chunky purée forms. Transfer purée to a saucepan and bring to a boil over medium heat, stirring constantly. Do not allow purée to scorch.
Remove from heat and add bourbon, vanilla, cinnamon, and allspice. Pour into clean glass jars and store in refrigerator. Consume within 2 weeks.