Whisk together the flour, baking powder, baking soda, cornstarch, and salt in a large bowl.
Beat the softened butter, granulated sugar, and brown sugar until light and fluffy, about 2 to 3 minutes. Add the egg and vanilla and mix until combined, then stir in the sour cream.
Gradually add the dry ingredients and mix just until the flour disappears. Cover the dough and refrigerate for 1 hour.
While the dough chills, combine the brown sugar, flour, and cinnamon for the filling. Mix in the softened butter and vanilla until a thick paste forms. Refrigerate the filling for about 30 minutes, then divide it into small portions.
Preheat the oven to 350°F (180°C) and line baking sheets with parchment paper.
Scoop the cookie dough into portions and flatten each one slightly. Place a small amount of the brown sugar filling in the center, then fold the dough around it and seal completely.
Arrange the stuffed cookies on the prepared baking sheets, leaving about 2 inches between each cookie. Gently flatten them and refrigerate for another 15 to 20 minutes.
Bake for 10 to 12 minutes, until the edges are set and the centers are still soft and slightly puffy. Let the cookies cool on the baking sheet for 5 minutes before transferring them to a wire rack.
Whisk the powdered sugar, cinnamon, and milk together to make the glaze. Add corn syrup if using, then spread a small amount over each completely cooled cookie.
Allow the glaze to set for 15 to 20 minutes before serving.