I started by using a basic butter cookie recipe (I prefer butter cookies to sugar) and a round cookie cutter. Place the cookies on a baking sheet then insert a lolly-pop stick into each cookie before baking. If the sticks come through you can reinforce that area with extra cookie dough. Not to worry you won't see it once the cookies have icing.
To Ice the cookies I use a standard Royal Icing Recipe.
3 Tablespoons Meringue Powder
4 cups (about 1 lb.) confectioners sugar
6 tablespoons warm water
4 cups (about 1 lb.) confectioners sugar
6 tablespoons warm water
Mix all ingredients together and separate into 5 zip-loc bags. Add the specified colors to each bag. The Blue bag should be the bag with the most icing. You will need Blue (Grover's Body), White (Grover's eye's), Pink (Grover's nose), Black (Grover's eyes), and Red (Grover's mouth).
Once Grover is furry let the cookies dry for about 20 minutes.
I use the same technique to make the other Sesame Street characters.