Preheat the oven to 180°C (160°C fan/350°F). Lightly grease three 15 cm (6-inch) cake tins and line the base of each with a circle of parchment paper.
Place the milk and star anise in a saucepan. Heat, stirring frequently to prevent a skin from forming, until a few small bubbles start to form on the surface. Pour into a jar or measuring jug and add the chai tea bags. Let steep for 10 minutes, then discard the star anise and tea bags. Cool to room temperature before using.
In a medium bowl, combine the flour, baking powder, baking soda, salt, and chai spice mix ingredients.
In a large mixing bowl, or the bowl of a stand mixer fitted with the paddle attachment, cream the butter and brown sugar until light and fluffy (about 3-4 minutes).
Add the sour cream, oil, eggs, and vanilla, and mix until combined.
Alternate adding ⅓ of the dry ingredients with ⅓ of the steeped milk, mixing after each addition until just combined. Repeat with the remaining ingredients. Use a rubber spatula to scrape down the bottom and sides of the bowl to make sure everything is incorporated.
Divide the batter evenly between the three cake tins and bake for 30-35 minutes, or until a skewer inserted into the centre comes out clean.
Let the cakes cool in their tins for about 15 minutes, then turn them out onto a wire rack to cool completely. Meanwhile, prepare the buttercream.