To make the cake: Heat oven to 325º. Butter and flour a light-colored 10-inch fluted tube pan, making sure there are no clumps of butter or exposed areas in pan; set aside.
Sift together flour, baking powder, and salt into large bowl and set aside. In the bowl of a standing mixer with a paddle set on medium speed, cream 3 sticks of butter until fluffy, about 2 minutes. Add the light brown and white sugars and mix, stopping and scraping down the bowl with a rubber spatula occasionally, until lightened and smooth, about 5 minutes. Reduce speed to medium-low and add eggs, one at a time, beating for about 15 seconds after each addition and scraping down the bowl occasionally, until well blended. Do not overmix. Reduce speed to low and add the flour mixture and 11⁄2 cups milk alternately, in 3 batches, beginning and ending with flour; beat until smooth, about 3 minutes total.
Scrape the batter into the prepared pan and bake in the center of the oven until a wooden skewer inserted in the center comes out clean, about 1 hour and 10 minutes. Do not overbake. Let cake cool in the pan on a wire rack for at least 30 minutes. Loosen edges with a small knife, then turn onto rack to let cool completely.
To make the icing: Melt remaining stick of butter in a large heavy saucepan over medium heat. Add dark brown sugar and whisk constantly until mixture is bubbling and smooth and butter is completely incorporated, about 3 minutes. Remove from heat and whisk in remaining 1⁄2 cup milk. Let mixture cool. Stir in the vanilla, then gradually add the confectioners' sugar and stir until well blended and smooth. Transfer cake to a cake plate and ice with frosting. Let stand for at least 2 hours before slicing.