Ugly Sweater Cake

Prepare the Cake: Preheat oven to 350°F. Grease a 13-by-9-inch baking pan with butter and dust with flour; set aside. Beat sugar and butter with a stand mixer fitted with a paddle attachment on medium speed until well combined, about 2 minutes. Add eggs 1 at a time, beating until just combined after each addition, about 1 minute total. Scrape down sides and bottom of bowl using a flexible spatula.

Sift together flour, cocoa, baking powder, baking soda, and salt in a separate bowl. Stir together milk, water, red velvet bakery emulsion, and vinegar in a 2-cup measuring cup or a medium bowl. With stand mixer running on medium-low speed, add flour mixture to butter mixture alternately with milk mixture, beating until just combined, about 1 minute. Transfer batter to prepared pan. Bake in preheated oven until a wooden pick inserted into center of cake comes out clean, 25 to 30 minutes. Let cake cool in pan 10 minutes. Run an offset spatula around edges of pan, and remove cake. Transfer cake to a wire rack, and let cool completely, about 1 hour.

Prepare the Frosting: Fill a medium saucepan with water to a depth of 1 inch; bring to a boil over medium-high. Reduce heat to low to maintain a gentle simmer. Place white chocolate in a medium-size heatproof bowl (bowl should be large enough to sit inside rim of saucepan but not touch the water below), and place bowl over simmering water. Cook, stirring often, until white chocolate is melted and only a few small lumps remain, about 3 minutes. Remove bowl from saucepan. Stir white chocolate constantly until fully melted. Let cool slightly, about 5 minutes.

Beat cooled melted white chocolate, cream cheese, butter, and vanilla in a large bowl with a stand mixer fitted with a paddle attachment on low speed until smooth, about 1 minute. Reduce mixer speed to low, and gradually beat in powdered sugar until combined, about 3 minutes. Measure ⅓ cup frosting into a small bowl; stir in green food coloring until desired shade of green. Spoon green-dyed frosting into a piping bag fitted with a star tip (such as Ateco 822). Reserve remaining undyed frosting in large bowl.

Place cake on a platter. Spread cake top and sides with an even, smooth layer of undyed white frosting. Arrange cake lengthwise in front of you (with 1 short edge facing you). Pipe green-dyed Frosting into a large "V" on the top third of the cake, with each top corner of the "V" meeting 1 top corner of the cake. Pipe a vertical line running lengthwise down the center of the cake starting at the bottom tip of the "V".  Pipe 2 horizontal lines running crosswise across the cake, 1 line about halfway down and 1 line about three-quarters of the way down the length of the cake. Arrange peppermint candies along the lengthwise, vertical line to make buttons. Fill the different sections of the cake "sweater" with assorted sprinkles.