Preheat the oven to 300°F. Place the peppermint candies in concentric circles in an 11-inch springform pan, starting with the green peppermints. Transfer the springform pan to the oven and melt the candies at 300°F for 15 minutes. Once they've melted, remove the springform pan from the oven and let the peppermint platter cool for 2 hours.
Beat together the cream cheese, powdered sugar, and whipped cream stabilizer or corn starch, then carefully fold in the whipped cream. For the best results, transfer the cream cheese frosting to the refrigerator and chill for 1–2 hours to make it easier to spread it on the cake later.
Preheat the oven to 350°F and grease the 7-inch cake pan. Beat the egg whites and gradually add in the sugar until a stiff meringue forms. Stir together the flour and cocoa powder, then add the dry ingredients, vegetable oil, and milk to the meringue, one ingredient at a time. Pour the cake batter into the cake pan, transfer the cake pan to the oven, and bake the cake at 350°F for 30–40 minutes.
Once the cake is done baking and has cooled slightly, remove the peppermint platter from the springform pan and place it on your work surface. Slice the chocolate cake in half lengthwise, creating 2 even layers, then place 1 half in the center of the peppermint platter. Transfer the cream cheese frosting to a piping bag and pipe a layer of it onto the first layer of chocolate cake. Crush 3 red and 3 green peppermints and sprinkle them over the layer of frosting.
Place the other half of the chocolate cake on top, then cover the entire cake with cream cheese frosting and smooth it out using a rubber spatula.
Heat the cream in a saucepan, then stir in the white chocolate chips and the red powder food coloring. Let the sauce cool briefly.
Repeat the same process using green powder food coloring for the green chocolate sauce, then let the sauce cool briefly.
Cut off the bottom of an empty plastic bottle and clean it thoroughly. Place it in the center of the Christmas cake so that the base is facing upwards, then use a wooden spoon to press 5 evenly spaced lines into the cream cheese frosting, lining up the lines with the grooves in the bottom of the plastic bottle. Slowly pour the green chocolate sauce onto the center of the bottom of the plastic bottle so that the grooves fill with the sauce and flow down into the lines on the cake. Rotate the water bottle and use the wooden spoon to press 5 more lines into the cream cheese frosting between the ones filled with green chocolate sauce. Pour the red chocolate sauce onto the center of the empty plastic water bottle so that the red sauce flows down like the green sauce did. Remove the bottom of the plastic bottle and transfer the remainder of the cream cheese frosting to a piping bag with a star-shaped nozzle. Pipe several swirls of frosting onto the top of the cake and decorate it with some additional peppermint mints.