Brown the butter. Transfer to a large mixing bowl and chill for 30 minutes, or until solidified but still soft.
Preheat the oven to 350°F. Prep an 8x8 or 9x9 inch pan by greasing it and lining it with parchment paper.
Cream together the brown butter, granulated sugar, and brown sugar for 2-3 minutes, scraping down the sides of the bowl as needed.
Mix in the eggs and vanilla, stirring well.
In a liquid measuring cup or small bowl, combine the milk and chai concentrate.
In a separate bowl, combine the flour, baking powder, salt, cinnamon, ginger, nutmeg, and cloves. Stir until the spices are evenly distributed.
Add half of the chai mixture to the sugar mixture, stirring to combine. Follow with half of the flour. Repeat with the remaining chai and the remaining flour.
Pour the batter into the prepared cake pan and bake at 350°F until a tester inserted into the center of the cake comes out clean. Using an 8x8-inch pan, bake for about 35-40 minutes. Plan on about 30-35 minutes for a 9x9-inch pan.
Set the cake aside to cool completely before frosting.