Peel peaches and cut into one-inch cubes. In a bowl, toss peaches with lemon juice and two tablespoons brandy or bourbon.

Cook sugars in a heavy skillet over medium heat, stirring slowly with a fork until they melt and turn a golden amber color.

Slowly add peaches and cinnamon (caramel will harden) and cook, stirring frequently, until caramel is dissolved and peaches are tender, about five minutes.

Transfer peaches to a bowl and cool to room temperature. Stir in remaining brandy or bourbon.

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.