Preheat the oven to 350°F (177°C). Line an 8-inch square baking pan with parchment paper and lightly grease it.
In a medium bowl, mix brown sugar, flour, and cinnamon. Add cold butter cubes and use a pastry cutter or hands to form clumps. Chill in the fridge.
In a large bowl, whisk together the flour, baking powder, baking soda, and salt. Set aside.
In another bowl, beat the softened butter and granulated sugar on high for about 3 minutes until light and creamy.
Add eggs and vanilla extract. Beat until smooth. Mix in sour cream. It may look slightly lumpy.
With mixer on low, gradually add dry ingredients to wet. Mix just until combined. Add milk and gently mix again.
Spread half the batter into prepared pan. Top with 1 cup of crumb mixture. Add remaining batter and spread gently. Top with remaining crumb.
Bake for 35–40 minutes. Check with a toothpick in the center. Cool for 15 minutes on a wire rack.
Mix icing ingredients until smooth. Drizzle over warm cake.
Lift cake from pan using parchment, slice, and serve warm or at room temperature.