Preheat your oven to 350°F (175°C). Line two large baking sheets with parchment paper.

In a small bowl, combine the well-drained diced peaches, 2 tablespoons light brown sugar, 1 teaspoon cornstarch, and 0.25 teaspoon ground cinnamon. Stir gently to coat the peaches, then set aside.

In a medium bowl, whisk together 0.25 cup all-purpose flour, 0.25 cup old-fashioned rolled oats, 0.25 cup light brown sugar, and 0.25 teaspoon ground cinnamon. Cut in the 0.25 cup cold, diced unsalted butter using a pastry blender or your fingertips until the mixture resembles coarse crumbs. Place in the fridge while you make the dough.

In a large bowl with an electric mixer, cream together the 0.75 cup softened unsalted butter, 0.5 cup granulated sugar, and 0.25 cup light brown sugar on medium speed until light and fluffy, about 2-3 minutes. Beat in the large egg and 1 teaspoon vanilla extract until well combined, scraping down the sides of the bowl as needed.

In a separate medium bowl, whisk together 1.5 cups all-purpose flour, 0.5 teaspoon baking soda, and 0.25 teaspoon salt. Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined. Be careful not to overmix.

Scoop about 1.5 tablespoons of cookie dough for each cookie onto the prepared baking sheets, spacing them about 2 inches apart. Use the back of a spoon or your thumb to create a shallow indentation in the center of each dough ball.

Spoon about 0.5-1 teaspoon of the peach filling into the center of each indentation. Sprinkle a generous amount of the prepared crumble topping over the peach filling.

Bake for 12-15 minutes, or until the edges are lightly golden and the centers are just set. The peach filling might bubble slightly. Let the cookies cool on the baking sheets for 5 minutes before transferring them to a wire rack to cool completely.