1. Peel peaches and cut into one-inch cubes. In a bowl, toss peaches with lemon juice and two tablespoons brandy or bourbon.
2. Cook sugars in a heavy skillet over medium heat, stirring slowly with a fork until they melt and turn a golden amber color.
3. Slowly add peaches and cinnamon (caramel will harden) and cook, stirring frequently, until caramel is dissolved and peaches are tender, about five minutes.
4. Transfer peaches to a bowl and cool to room temperature. Stir in remaining brandy or bourbon.
5. Spoon two tablespoons of peach mixture into each glass and top with a small scoop of ice cream. Repeat with another layer of peaches and ice cream and serve immediately.