Preheat oven to 350°F. Grease and line a 9×9‑inch light-colored metal pan with parchment.

In a bowl, whisk together the flour, cinnamon, ginger, allspice, nutmeg, cardamom, cloves, baking powder, baking soda, and salt. 

In another bowl, cream softened butter with granulated and brown sugar on high speed for 2 minutes. Add eggs and vanilla; mix until pale and smooth (~1 minute).

Alternately add buttermilk and dry ingredients on low speed until just combined and smooth. Scrape bowl as needed. 

Pour batter into prepared pan and bake 35–38 minutes, or until a toothpick inserted in center comes out clean. Let cool in pan for 30 minutes before removing via parchment to finish cooling. 

Meanwhile, make the brown butter frosting: melt butter in saucepan over medium; simmer until foamy with a nutty aroma and deep golden brown (about 10–12 minutes). Chill until soft but solid. Whip on high until pale and fluffy (5–10 minutes), add cold cream cheese and whip until fluffy. Add powdered sugar, mix on low until blended, then on high (~1 minute) until light. 

Make the chai milk soak: heat milk until steaming, add teabags, steep 20 minutes, remove teabags and let cool. Mix in condensed milk and vanilla, set aside.

Once cake is completely cool, slice off a thin top layer, poke holes across the surface, then slowly pour chai milk mixture over, letting it soak in. 

Frost with thick layer of brown butter cream cheese frosting, dust with cinnamon if desired, slice into 16 squares, and serve.