Preheat the oven to 350°F. Grease and flour a 9x13-in. baking pan. Set aside.
In a small saucepan, warm the milk just until it simmers. Remove from the heat and add the tea bags. Allow to steep for 10 minutes. Remove the tea bags and allow the milk to cool to room temperature.
In a medium bowl, combine the flour, baking soda, baking powder, salt, cinnamon, cardamom, ginger, nutmeg and cloves. Set aside.
In a large bowl, cream the butter and sugar together. Add the eggs, one at a time, beating after each addition until it is combined. Scrape the bowl down, then add the applesauce. Mix just until incorporated.
Add half of the flour mixture to the batter and mix just until combined. Add half of the chai milk, mixing on low just until combined. Add the other half of the flour mixture, beating just until combined. Scrape the bowl down and add the remaining milk. Mix until fully combined - do not over mix.
Bake for 25-30 minutes or until the cakes tests done when a toothpick is inserted near the center. Remove from the oven. Allow to cool slightly.
Meanwhile, in a small saucepan, warm the 2 cups of milk just until it simmers. Remove from the heat and add the tea bags. Allow to steep for 10 minutes. Remove the tea bags and allow the milk to cool to room temperature. Once cool, put the pudding mix in a medium bowl, add the chai milk and whisk until smooth.
Poke holes all over the cake with the handle end of a wooden spoon. There is no set number but you want plenty of holes for the pudding to fill. Pour the pudding mixture over the cake, trying to fill the holes the best you can. You may need to use the back of a spoon to help push the pudding into the holes. Evenly spread the Cool Whip over the cake.
Refrigerate at least 4 hours before serving. Store and leftovers, covered, in the refrigerator.