Spray and line an 8 x 4-inch loaf pan with parchment paper. (Note: If using a 9 x 5-inch loaf pan, reduce the baking time slightly.)
Heat the butter in a small pot, stirring occasionally, until browned bits form on the bottom of the pot.
Transfer the butter to a heatproof measuring cup. Set aside.
In a medium bowl, whisk to combine the oil, sugar, brown sugar, eggs, 1 ½ teaspoons salt, cinnamon, and ginger.
Fold in the flour, baking soda, and brown butter, reserving ¼ cup brown butter in the measuring cup to use later for the icing.
Fold in the pineapple, carrots, ½ cup pecans, and coconut to combine.
Pour the batter into the prepared loaf pan.
Bake for 60-65 minutes or until a wooden pick inserted in the center comes out mostly clean with a few moist bits. Set the cake onto a cooling rack to cool at least 30 minutes.
Unmold the cake and remove the parchment paper. Cool completely.
To prepare the icing, beat the reserved ¼ cup brown butter with the cream cheese until uniform.
Add the confectioners’ sugar, vanilla, and remaining ¼ teaspoon salt and beat until smooth.
Transfer to a piping bag and pipe (or spoon) over the cooled pound cake. Sprinkle with the remaining ¼ cup nuts.