Preheat your oven to 350°F (175°C). Grease a 9x9 inch baking pan and line with parchment paper.

In a medium bowl, whisk together all-purpose flour, cinnamon, ginger, allspice, nutmeg, cardamom, cloves, baking powder, baking soda, and salt. Set aside.

In a large mixing bowl, cream together the softened unsalted butter, granulated sugar, and brown sugar until fluffy. Beat in eggs and vanilla extract until smooth.

Gradually add buttermilk into the butter mixture, alternating with the dry ingredients. Mix at low speed just until combined.

Pour the batter into the prepared baking pan and bake for 35-38 minutes, or until a toothpick comes out clean.

Let the cake cool in the pan for about 30 minutes, then transfer to a wire rack to cool completely.

Brown the unsalted butter in a saucepan over medium heat until golden. In a mixing bowl, beat browned butter with cream cheese and powdered sugar until smooth.

In a small saucepan, combine whole milk and chai teabags over low heat, steep for about 10 minutes. Mix in sweetened condensed milk and vanilla extract.

Poke holes in the cooled cake, then pour the chai milk soak evenly over it.

Spread the brown butter cream cheese frosting over the top of the chai cake. Sprinkle with ground cinnamon before serving.