Knead together all of the cookie dough ingredients by hand, making sure to mix everything quickly so that the sugar doesn't dissolve completely. Divide the dough in half, then color one half red and the other half green with the red & green gel food coloring. Roll both halves of the dough into balls, wrap them in plastic wrap, transfer them to the refrigerator, and chill for 45 minutes.
Once the dough has chilled, use an apple corer to cut the two balls of cookie dough so that you create 8 equally sized segments and 1 cylindrical piece of dough in each color.
Piece the segments of dough back together, alternating between the red & green pieces, so that you're left with 2 patterned balls of cookie dough. Gently squeeze the balls of red & green dough together, transfer them to the refrigerator, and let them chill for 30 minutes.
Preheat the oven to 350°F and line 2 baking trays with parchment paper. Place 1 sheet of parchment paper over each of the 2 balls of cookie dough, then carefully press down on both of them with a baking tray or large plate, creating 2 giant cookies, each about 7 inches in diameter. Use a knife to even out the edges if necessary, then place both cookies on baking trays. Transfer the baking trays to the oven and bake at 350°F for 20 minutes.
Melt the cream cheese, white chocolate chips, vanilla extract, cinnamon, and lemon juice over a double boiler and stir everything together. Once everything is thoroughly combined, remove the cheesecake filling from the heat and let it cool briefly. Whip the butter until light and airy, then fold it into the cheesecake mixture. Spread the majority of the cheesecake filling over 1 of the Christmas cookies, making sure to save some for later.
Break off pieces of the gingerbread man and carefully press the pieces into the cheesecake filling. Place the second Christmas cookie on top of the cheesecake filling.
Transfer the rest of the cheesecake filling to a piping bag with a star-shaped nozzle, then pipe small amounts of it on top of the cake. Insert 1 gold sugar pearl into each portion of cheesecake filling on top of the Christmas cookie cake.
