Preheat your oven to 350°F (175°C) and line a large baking sheet with parchment paper.

In a large mixing bowl, whisk together the melted vegan butter, light brown sugar, granulated sugar, non-dairy milk, cornstarch, vanilla extract, cinnamon, baking soda, and kosher salt until the mixture is smooth and well combined. Using a rubber spatula or wooden spoon, gently fold in the all-purpose flour until a soft, cohesive dough forms.

Sprinkle about ½ teaspoon of brown sugar onto the prepared baking sheet in several spots to create a base for each cookie. Lay 4–5 thin peach slices over each sugar mound. Then, scoop approximately 1.5 tablespoons of cookie dough and place it directly on top of the peaches. Gently press the dough down with the palm of your hand to help the peaches adhere.

Repeat this process with the remaining dough and peaches, spacing the cookies apart to allow for spreading. Bake for 12–13 minutes, or until the edges are golden and the centers are slightly soft.

While the cookies are still warm, take a large round cookie cutter or the rim of a glass and gently swirl it around each cookie to shape them into perfect circles. Let the cookies cool on the baking sheet for 10 minutes, then transfer them to a wire rack to cool completely.