Preheat oven to 350 degrees F. Line a baking sheet with parchment paper.
Slice cookie dough into 12 equal rounds. Place on the prepared baking sheet and bake 8 to 9 minutes -- slightly underdone with pale golden edges. They will firm up as they cool. Cool completely on the pan.
Combine diced peach, sugar, vanilla, and cinnamon in a small bowl. Let sit 10 minutes until peaches release their juices and mixture becomes slightly syrupy.
In a bowl, fold the macerated peach mixture into the softened ice cream until evenly distributed.
Scoop a generous portion of the peach ice cream onto the flat side of one cooled cookie. Press a second cookie flat-side down on top. Wrap each sandwich in plastic wrap or parchment.
Freeze at least 1 to 2 hours until firm before serving.