Preheat your oven to 350°F (175°C) and line a baking sheet with parchment paper.
Beat the softened butter, granulated sugar, and brown sugar together until light and fluffy, about 2-3 minutes with an electric mixer.
Mix in the eggs one at a time, then stir in the vanilla extract until well combined.
Whisk together the flour, baking soda, and salt in a separate bowl. Gradually add the dry ingredients to the wet mixture, mixing until just combined.
Gently fold in the sliced peaches, making sure they stay in big chunks without breaking apart.
Scoop spoonfuls of dough onto the baking sheet, spacing them about 2 inches apart. Lightly sprinkle cinnamon sugar on top of each cookie.
Bake for 12-15 minutes, or until the edges are golden brown and the centers are set.
Let the cookies cool on the baking sheet for a few minutes before transferring them to a wire rack to cool completely.