Preheat oven to 350°F (177°C). Line an 8-inch square pan with lightly greased parchment paper or directly grease the pan.
Combine the brown sugar, flour, and cinnamon together in a medium bowl. Cut in the chilled butter until the mixture is in pea-sized crumbs. Refrigerate until step 5.
Whisk the flour, baking powder, baking soda, and salt together in a large bowl. Set aside. In a large bowl, beat the butter and sugar together on high speed until smooth and creamy, about 3 minutes. Add the eggs and vanilla, and beat on medium-high speed until combined. Beat in the sour cream.
Add the flour mixture to the wet ingredients, and beat on low speed until just combined. Finally, beat in the milk. The batter will be thick.
Spread about half of the batter into the prepared pan. Sprinkle 1 cup of the crumb mixture evenly on top. Spread the remaining batter on top. Sprinkle the remaining crumb mixture evenly on top.
Bake for around 35–40 minutes or until the cake is baked through. Allow cake to cool in the pan set on a wire rack for 15 minutes.
Whisk all of the icing ingredients together. Drizzle over warm cake.
Lift the cake from the pan using the overhang parchment paper around the sides and slice into squares. Serve cake warm or at room temperature.
