Cut the butter into even pieces and add to a light-colored saucepan.

Cook over medium heat, stirring constantly, until the butter turns deeply golden and smells nutty.

Immediately pour into a heatproof measuring cup, scraping in all the brown bits.

Let cool for about 10 minutes. The butter should feel warm, not hot, like bath water.

In a bowl, combine the brown sugar, flour, cinnamon, vanilla and salt till well combined. Add the browned butter used for the streusel and pour over the mixture.

With a spoon or clean hands form both small and large buttery clumps.

Add the walnuts till combined.

Chill in the refrigerator to set while making the batter.

Preheat oven to 350°F (175°C). Line an 8×8-inch square pan with parchment.

Whisk flour, baking powder, baking soda, salt, and cinnamon. Set aside.

In another bowl, whisk the cooled brown butter used for the cake batter and add with both sugars. Mix till combined, it will look like wet sand.

Add eggs one at a time, then vanilla.

Whisk in sour cream.

Fold in dry ingredients just until combined. Do not overmix.

Gently add in shredded carrots until combined.

Spread the batter into a lined 8-inch square pan. Top with the chilled brown butter cinnamon streusel.

Bake at 350°F for 35–40 minutes, until the center springs back lightly and a toothpick inserted shows moist crumbs but no wet batter.

Allow the cake to cool in the pan for 20 minutes, then transfer to a wire rack to cool completely.

While the cake is baking, whisk together the softened cream cheese and pwodered till smoth a paste like consistency is formed.

Add milk, vanilla, and a pinch of salt, mixing until creamy.

Add more milk for a thinner drizzle consistency.

Pour over the cooled carrot coffee cake.