Step 1: In a large bowl, cream together 1 cup softened unsalted butter and 1 ½ cups granulated sugar until light and fluffy. Beat in the large egg and 2 teaspoons vanilla extract until well combined.
Step 2: In a separate medium bowl, whisk together 2 ¾ cups all-purpose flour, 1 teaspoon baking soda, ½ teaspoon baking powder, and ½ teaspoon salt.
Step 3: Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined. Stir in 2 tablespoons of milk.
Step 4: Divide the dough into 5 equal portions. Add a different gel food coloring (pink, yellow, green, blue, purple) to each portion, kneading gently until the color is evenly distributed. Wrap each colored dough portion in plastic wrap and chill for at least 30 minutes.
Step 5: Preheat oven to 375°F (190°C). On a lightly floured surface, roll out each chilled dough portion to about ¼-inch thickness. Use a small round or flower-shaped cookie cutter (about 2 inches) to cut out cookies. Reroll scraps as needed.
Step 6: Place the cut cookies about 1 inch apart on baking sheets lined with parchment paper. Bake for 8-10 minutes, or until the edges are lightly golden. Let cookies cool on the baking sheet for a few minutes before transferring to a wire rack to cool completely.
Step 7: While cookies cool, prepare the vanilla filling. In a medium bowl, beat ½ cup softened unsalted butter until creamy. Gradually add 2 cups powdered sugar, beating until smooth. Mix in 2 tablespoons heavy cream (or milk), 1 teaspoon vanilla extract, and a pinch of salt. Beat until light and fluffy.
Step 8: Once the cookies are completely cool, match them into pairs of similar sizes and colors. Spread or pipe a generous amount of vanilla filling onto the flat side of one cookie, then gently top with its matching partner to create a sandwich. Repeat with the remaining cookies.
Step 9: Serve immediately or store in an airtight container.