Preheat your oven to 350°F (175°C). Prepare a mini round cake pan or similar by setting it aside.
In a large mixing bowl, cream the butter and granulated sugar together until light and fluffy.
Add in the eggs, vanilla extract, and sour cream. Mix until well combined.
In a separate bowl, whisk together the flour, baking powder, cornstarch, and salt. Gradually add the dry ingredients to the wet ingredients, mixing until the cookie dough just comes together. Set aside.
In another bowl, mix together the melted butter and brown sugar until fully combined.
Add about a tablespoon of the brown sugar mixture to the bottom of each mini cake pan cavity, spreading it evenly.
Place a pineapple slice on top of the brown sugar mixture and position a maraschino cherry in the center of each pineapple.
Using a 3-tablespoon cookie scoop, portion the cookie dough and place it on top of each pineapple slice. Use an offset spatula to flatten the dough evenly.
Bake in the preheated oven for 25 minutes, or until the edges of the cookies are golden brown and the brown sugar mixture is bubbling around the sides.
Allow the cookies to cool in the pan for about 10 minutes. Then, carefully turn them out onto a wax paper-covered surface (to catch any excess brown sugar that may spill out). If any pineapples stick to the pan, gently reposition them on the cookies.
Let the cookies cool completely before serving. Store them in an airtight container. For the best flavor and texture, enjoy these cookies within 2-3 days.