Melt the butter in a medium skillet over medium heat until it begins to brown. Add the brown sugar, peaches and a pinch of cinnamon and cook, stirring occasionally, until the peaches are soft, about 10 minutes. Cool completely.
Put the ice cream and milk in a blender; add the peach mixture and blend until smooth. Transfer to mason jars or tall glasses. Top each with some crumbled cookies and a pinch of cinnamon. Serve immediately.