Set your oven to 350°F (175°C). Line a baking sheet with parchment paper.

In a large mixing bowl, beat softened butter and granulated sugar until light and fluffy, about 2 minutes. Add the egg and vanilla extract, mixing until combined.

In another bowl, whisk together all-purpose flour, baking soda, baking powder, and salt. Gradually incorporate this dry mix into the wet ingredients on low speed, forming a thick but soft dough. Add the milk and mix just until combined.

Divide the dough into five portions. Tint each portion with a different gel food coloring (pink, yellow, green, blue, and purple), kneading until the color is evenly distributed.

Roll tablespoon-sized portions into smooth balls and place them 2 inches apart on the prepared baking sheet. Bake for 8-10 minutes until edges are set and tops show slight cracks but no browning. Let cool on the baking sheet for 5 minutes, then transfer to wire racks to cool completely.

Beat softened butter in a bowl until smooth. Gradually add powdered sugar, then mix in heavy cream (or milk), vanilla extract, and a pinch of salt. Continue beating until fluffy and spreadable, adjusting thickness as needed.

Pair cooled cookies of similar size. Spread a generous amount of filling on the flat side of one cookie, then top with the matching cookie. Press gently to spread the filling.

Allow the assembled cookie sandwiches to rest for about 10 minutes to firm up the filling.