Preheat the oven to 350 degrees, and spray a bundt pan generously with cooking spray. Set aside.
Prepare an ice bath. In a small, stainless steel pot, heat the butter over low heat until melted. Increase the heat to medium and cook the butter until the milk solids turn golden brown and the butter has a nutty aroma. Place in the ice bath to cool.
Meanwhile, whisk together the flour, baking powder, baking soda and salt in a small bowl. Set aside.
Place the brown sugar and granulated sugar in a large bowl. When the butter has solidified around the edges and is no longer hot, remove from the pan with a rubber scraper and place in the bowl with the sugars. Beat on medium high speed until light and fluffy, about 2-3 minutes.
Add the eggs, one at a time, beating on low speed until incorporated.
Add the banana, milk, pineapple, coconut, and vanilla, and beat on low until combined.
Fold in the dry ingredients just until incorporated, then pour into the prepared bundt pan and spread as evenly as possible.
Bake on the middle rack of the oven for 40-45 minutes, or until the cake is golden brown and a toothpick inserted into the center comes out clean. Cool for about 30 minutes on a wire rack, then run a knife around the edges and center of the cake carefully, and invert onto the wire rack to cool completely out of the pan.