Place the chai tea bag into a mason jar. In a small saucepan over medium-high heat, warm the whole milk, stirring constantly, until it begins to boil. Pour the boiling milk over the chai tea bag in the mason jar and let steep for 20 minutes before removing the tea bag. Allow the mixture to cool completely to room temperature before moving on.
Preheat the oven to 350ºF and prepare three 6-inch or two 9-inch cake pans by spraying the sides with cooking spray and placing a wax or parchment paper circle into the bottom of each one.
Sift the cake flour and then measure by spooning into your measuring cup and leveling it or weighing with a food scale. Place it into a medium sized bowl and add the baking powder, baking soda, salt, and chai spice mix. Whisk together and set aside.
In the bowl of your stand mixer (a handheld mixer works fine too!), beat the butter on high for two minutes until it's light and creamy. Add in white and brown sugars and continue to mix on high for another two minutes, scraping down the bowl and paddle as needed. Add the eggs one at a time, mixing thoroughly after each addition. Add pumpkin puree and vanilla, turn the mixer to medium-high, and beat for one minute. It will look slightly curdled at this point but don't worry – it will come together as a smooth cake batter by the end of the recipe.
With the mixer on low speed, add in the dry ingredients and mix until just combined. Add the (room temperature!) chai milk mixture in a steady stream and mix until just incorporated. Scrape down the sides and bottom of the bowl and give it a few stirs to make sure there are no lumps. The batter will be slightly thick.
Pour the batter evenly between the cake pans (fill them no more than ⅔ full) and bake for 35-40 minutes, until a toothpick inserted comes out clean. Cool the cakes in the pan for ten minutes before removing and continuing to cool on a wire rack or flat surface.