Bring the water, butter, sugar, and salt to a boil in a saucepan over medium heat. Add the flour and stir continuously until a smooth dough forms. Remove the saucepan from the heat, let the mixture cool, and then stir in the eggs. Transfer the choux pastry dough to a piping bag, pipe 4 large spirals on a baking tray lined with parchment paper, and freeze for 1 hour.
Fry the churro spirals in hot cooking oil until golden brown, coat them in cinnamon sugar while still warm, and then let cool.
Cut off the top of the pineapple and use a glass to remove the inner core. Place the cinnamon sticks inside the pineapple, transfer the pineapple to a preheated oven, and cook at 325°F for 40 minutes. Let the pineapple cool for 30 minutes and then use the cinnamon sticks to scoop out cooked cinnamon pineapple chunks.
Beat the cream cheese with a hand mixer until smooth and then fold in the whipped cream and key lime zest.
Place the first churro spiral inside a tall 10-inch springform cake pan and top with a layer of key lime cream and a layer of cinnamon pineapple chunks. Place the second churro spiral on top and continue layering in this order until you reach the last churro spiral. Top the final churro spiral with a final layer of key lime cream and cinnamon pineapple chunks and garnish the churro cake with key lime zest and chocolate sauce before serving.