1. Preheat the oven to 350 degrees F. Butter the sides and bottom of a 9×13-inch glass or light-colored baking pan. Line the pan with parchment paper.
2. In a medium bowl, whisk the flour and salt together.
3. Put the white chocolate and butter in a large bowl and set it over a saucepan of simmering water, stirring occasionally, until the chocolate and butter are completely melted and smooth. Turn off the heat, but keep the bowl over the water and add the sugars. Whisk until completely combined, then remove the bowl from the pan. The mixture should be room temperature.
4. Add 3 eggs to the white chocolate mixture and whisk until combined. Add the remaining eggs and whisk until combined. Add the vanilla and stir until combined. Do not overbeat the batter at this stage or your brownies will be cakey.
5. Sprinkle the flour mixture and the coconut over the chocolate mixture. Using a rubber spatula (not a whisk), fold the flour and coconut mixture into the chocolate until just a bit of the flour mixture is visible.
6. Pour the batter into the prepared pan and smooth the top. Bake in the center of the oven for 30 minutes, rotating the pan halfway through the baking time, until a toothpick inserted into the center of the brownies comes out with a few moist crumbs sticking to it. Let the brownies cool completely, then lift them out of the pan using the parchment paper. Cut into squares and serve.
7. Store at room temperature in an airtight container or wrap with plastic wrap for up to 3 days.