Preheat oven to 350 degrees F. Grease the bottom and sides of 2 (8-inch) round cake pans; set aside.
In a large mixing bowl, combine cake mix, water, oil, eggs and extract. Use an electric mixer to beat on low speed for 1 minute. Scrape down bowl and then beat on medium speed for 2 minutes, or until batter is smooth. Divide batter evenly between cake pans.
Bake in preheated oven for 25 to 35 minutes, or until center of tester comes out clean.
Let cool in pans for 5 minutes. Remove from pans and transfer to a wire rack to cool completely.
For Royal Icing:
With a hand mixer, beat the egg whites with the lemon juice or extract until frothy. Add the sifted powdered sugar and beat on low speed until combined, smooth, and shiny. Turn to high and beat approximately 5 minutes until stiff and glossy. Add food coloring, if desired, and transfer to a pastry bag to pipe onto cookies.
Line the backside of a sheet pan with aluminum foil. Place 4 inch wide strips of parchment paper around the border of the sheet pan to keep the foil clean while you decorate the cake. Place cake rounds side by side on the lined sheet tray. Slice one side off each cupcake. Place cupcakes side by side at one end of a cake round with the cut/flat side facing up (this will be the top of the top hat).
Frost both cakes and cupcakes with frosting so that it looks like a hat is resting on a snowman's head resting on a body. Remove the parchment paper.
Spread or drizzle the bottom third of the sheet pan with some royal icing (the royal icing will act as glue for the coconut). Sprinkle the coconut over the royal icing to mimic snow. Randomly place 20 dollops of royal icing around the rest of the sheet pan. Place the snowflake cookies on each dollop of royal icing.
Decorate the snowman's hat with rows of blue mints, starting at the very top of the Snowman's head and working up onto the cupcakes. To decorate the face use the peppermint patties for eyes, place a small dot of the royal icing in the center of each patty and place a blue jellybean on top of the royal icing. Use the orange gum drop for the nose and the blue candy coated chocolates for the smile. To make the buttons dip the flat side of a hard candy into the royal icing and then into the colored decorating sugar and place on the snowman's chest; repeat with remaining 2 hard candies.
Take 2 (8-inch) piece fruit strips and tie them into a knot. Drape fruit strips across where the 2 cake rounds meet to make a scarf.
To make the broom , take a 3-inch long piece of fruit strip and slice the bottom 2 inches into thin strips. Place a small amount of royal icing at the end of the over sized candy cane and place the unsliced end of the fruit strip on to the royal icing.
Use the large white gumdrops to make the hands. Place the candy cane next too one of his hands.