BUTTER AND SUGARS: In a large bowl, use a hand mixer to beat together the butter, brown sugar, and white sugar. (The butter should not be melted at all. Beat until ingredients are completely smooth and combined, and light and creamy.)
FINISH WET INGREDIENTS: Add in the egg and vanilla extract and beat until combined.
DRY INGREDIENTS: Stir together the baking soda, salt, unprepared pudding mix, unprepared cake mix, flour, and nonpareil sprinkles in a separate bowl. (See Note 1.) Stir until combined.
COMBINE: Add the dry ingredients to wet. Mix gently (otherwise sprinkles begin to bleed colors) until just combined; do not overmix. Heads-up: this cookie dough gets thick!
SHAPE COOKIE DOUGH BALLS: Form cookie dough balls that are each 2 tablespoons in size. Roll balls that are taller rather than wider. Place cookie dough balls on a plate and cover. Refrigerate for at least 1 hour and up to 2 hours.
BAKE: Preheat the oven to 350 degrees F. Line a baking sheet with parchment paper or a baking liner. Place no more than 6 cookie dough balls per cookie sheet to give ample room for the cookies to spread. Bake for 9-13 minutes or until lightly browned at the edges and still soft (slightly under-baked in the center-- we bake for about 11 minutes). Remove from the oven and, working quickly, immediately press the edges of each cookie slightly inwards with the back of a spoon.
LET STAND: Remove from the oven and allow cookies to stand on the cookie sheet for another 3-4 minutes. Remove with a metal spatula to a cooling rack until fully cooled.
FROSTING: Meanwhile, prepare the frosting. In a large bowl, beat the room-temperature butter and cream cheese until completely smooth; mix in the vanilla and salt. Add in the powdered sugar and then beat until the frosting is smooth. It will seem like you need more liquid, but just keep beating; it will come together! If it does happen to be too wet, add a bit more powdered sugar and if it is too dry, add a touch of milk or cream.
DECORATE: Transfer the frosting to a plastic bag and cut off the tip (or piping tip fitted with a small tip. Frost each cooled cookie by piping the frosting in a swirl on top. Sprinkle tops of the cookies with extra sprinkles and press a frosted animal cracker into the center of each cookie.
STORE: Store any leftover (completely cooled) cookies in an airtight container at room temperature. Frosted cookies are best enjoyed the same day they're made.