Heat the oven to 350°F with a rack in the middle position. Mist a broiler-safe 9-by-13-inch baking pan with cooking spray, dust evenly with flour, then tap out the excess. In a small bowl, whisk together the flour and baking powder, then sift into a medium bowl or onto a large sheet of kitchen parchment.

In the bowl of a stand mixer with the whisk attachment, beat the eggs, white sugar, vanilla and salt on low until sugar dissolves, about 20 seconds. Increase to medium-high and whip until the mixture is pale and thick, 5 to 6 minutes.

Reduce to low, add about one-third of the flour mixture and mix until almost incorporated, about 10 seconds. With the mixer running, slowly pour in ½ cup of milk. Repeat with half of the remaining flour mixture and another ½ cup of the remaining milk.

With the mixer still running, add the remaining flour mixture and mix just until no flour clumps remain, about 20 seconds. Using a silicone spatula, fold the batter, scraping along the bottom of the bowl, to ensure the ingredients are well combined.

Pour the batter into the prepared pan. Bake until just beginning to brown and a toothpick inserted into the center of the cake comes out clean, about 22 minutes. Set the pan on a wire rack and heat the broiler.

In a medium saucepan, combine the remaining ¾ cup milk, the brown sugar and butter. Bring to a boil over medium-high, stirring to dissolve the sugar, then boil for 4 minutes. Remove from the heat, add the coconut and stir until evenly moistened.

Spoon the topping onto the outer edges of the warm cake, then, using an offset or silicone spatula, gently spread into an even layer over the surface. Broil until the top is bubbling and deep golden brown, 2 to 3 minutes, rotating the pan if needed for even browning.

Set the pan on a wire rack and tent with foil. Let cool for at least 30 minutes. Cut into pieces and serve warm or at room temperature.