Choose cutters with simple shapes; they're easier to handle and less likely to break. If you want to make small puzzles, cut dough into 4-inch squares, and use a single cookie cutter to punch out the center of each square while still hot from the oven. Working quickly, cut lines from edges of square to form puzzle pieces. Store icing in an airtight container at room temperature up to 1 week.
Beat first 2 ingredients at medium speed with an electric mixer until creamy. Add egg and vanilla; beat until blended.
Sift together flour and salt. Gradually add flour mixture to butter mixture, in 2 batches, beating on low speed until just combined after each addition. Flatten dough into 2 rectangles, and wrap in plastic wrap. Chill 2 hours to 3 days.
Preheat oven to 350°. Place dough on floured parchment paper, and roll to 1/4-inch thickness. Cut into a 7- x 7 1/2-inch rectangle, reserving scraps. Transfer parchment to a baking sheet, and bake 7 minutes. Rotate pan front to back, and bake 7 more minutes or until edges begin to brown.
Immediately slide parchment with cookie onto a flat surface. Cut through cooked, soft dough with a small cookie cutter. Cut straight lines from shapes to rectangle edges to form puzzle pieces. Transfer parchment paper with cookies to wire rack. Gently separate puzzle pieces. Cool completely. Repeat with remaining dough.
Beat powdered sugar, meringue powder, and 10 Tbsp. warm water at high speed 5 minutes or until glossy. Stir in up to 4 Tbsp. warm water, 1 tsp. at a time, until mixture reaches desired consistency. Stir in desired food coloring.
Pipe a thin border of icing around edge of each cookie piece. Fill center of cookie with an even layer of icing. Swirl a wooden pick in a circle around the entire pool of icing, starting at the edges and moving toward the center. Let cookies stand 30 minutes or until icing is dry before decorating with another color on top of base color.
For crisp cookies, return dough immediately to oven after cutting in Step 4, and bake 3 minutes or until golden.