Heat the oven to 350°F and arrange a rack in the middle. Coat a 13-by-9-inch baking dish with butter; set aside. Place the coconut in a single layer on a baking sheet and bake until toasted and golden brown in color, about 5 minutes; set aside. Place the eggs in a large bowl and lightly beat to break up the yolks. Add the remaining ingredients and whisk until smooth, about 2 minutes. Pour the mixture into the prepared dish and bake for 25 minutes. Sprinkle with the toasted coconut, rotate the dish, and bake until the edges are just starting to brown and the top is just set (a bubble may form, but it will flatten as the cake cools), about 20 to 25 minutes more. Remove from the oven and let cool for 30 minutes before serving. Wrap leftovers tightly with plastic wrap and refrigerate.