Preheat oven to 350 degrees Fahrenheit (175 degrees Celsius).
In the bowl of a stand mixer, cream butter and sugars. Add vanilla and egg, and mix until incorporated.
In a separate mixing bowl, sift flour, cornstarch, baking soda, and salt. Mix in dry ingredients, one cup at a time, until smooth.
Add M&M’s or chocolate chips, and gently fold in.
Place cookies on a sheet using a standard scoop. Bake for 8-10 minutes or until the edges are golden brown. (Since I made giant cookies, I used 2 scoops of dough, patted them together, put 6 on a sheet, flattened them, and baked them for 14 minutes).
Remove from oven and let cool on the sheet for 5 minutes before transferring to a wire rack to cool completely.
While cookies are cooling, prepare frosting by beating together butter, milk, salt, and extracts until light and fluffy. With mixing speed on low, slowly add the powdered sugar. Beat for 2-3 minutes. Pipe (or spread) onto cookies & sandwiches.
Notes
To STORE Chocolate Chip Cookie Sandwich, place them in an airtight container and keep them at room temperature for up to 3 days. For longer storage, you can FREEZE the sandwiches in a single layer on a baking sheet, then transfer them to a freezer-safe bag.They will stay good in the freezer for up to 2 months. To enjoy frozen cookie sandwiches, let them thaw at room temperature for about 15-20 minutes.