Bake the cake in a 9×13-inch pan according to package directions.
While the cake is still warm, poke holes all over the top using the handle of a wooden spoon or a straw.
In a bowl, whisk together the Baileys Churro and sweetened condensed milk. Slowly pour the mixture evenly over the cake, letting it soak into all those holes.
Refrigerate for at least 1 hour to let the flavors set.
In a chilled mixing bowl, beat the cold heavy cream on medium-high speed until it begins to thicken.
Add the powdered sugar, cinnamon, and vanilla extract. Continue whipping until soft peaks form.
Spread the cinnamon whipped cream over the chilled cake. Sprinkle generously with cinnamon sugar and finish with a drizzle of caramel sauce.