Heat oven to 350°F. In medium bowl, stir together cookie mix, butter and egg until dough forms a ball.
On floured surface with rolling pin, roll out dough to 1/4-inch thickness. Cut with round cookie cutter; place rounds about 1 inch apart on ungreased cookie sheets. (We used a 4" cookie cutter for this recipe).
Bake 9-10 minutes or until edges are golden brown. Remove from cookie sheets to cooling racks. Cool completely, about 15 minutes.
In bowl of electric mixer fitted with paddle attachment, beat water and meringue powder until foamy. Add powdered sugar, corn syrup and lemon extract; mix on low speed until combined, scraping side of bowl occasionally. Increase speed to medium-low; beat 5 minutes. Increase speed again to medium-high; beat until mixture forms stiff peaks. Transfer Royal Icing to airtight container.
Transfer 1 cup icing to bowl; stir in white food color until bright white. Spoon into decorating bag fitted with small round tip. Pipe outline on outer edge of each cookie. Let icing set 1 hour.
Transfer another portion of icing to bowl; stir in white food color until bright white. Stir in water, a small amount at a time (about 1/2 to 1 teaspoon), until well blended. Continue to stir in water until icing falls back into itself. Spoon thinned icing into decorating bag fitted with small round tip. Fill center of each cookie with icing. Use knife or toothpick to bring icing to edges. Let cookies set 6 hours or overnight.
Transfer 1 cup icing to bowl; stir in red food color. Spoon icing into decorating bag fitted with small round tip. Using a Wilton #3 icing tip or smaller, pipe baseball details on cookies. Let icing set until completely dry.