In the bowl of a stand mixer fitted with the dough hook attachment, combine warm coconut milk, sugar, melted coconut oil, salt, and egg. Add flour and yeast, and beat at low speed until a dough forms. Increase mixer speed to medium, and knead until smooth, about 8 minutes. Cover with plastic wrap, and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, about 1½ hours.
Preheat oven to 350°F (180°C). Spread coconut flakes on a baking sheet, and bake, stirring halfway through, until lightly golden brown, about 8 minutes. Transfer coconut to a bowl, and let cool completely. Lightly grease a 13x9-inch baking pan with butter.
Transfer dough to a lightly floured surface. Roll into an 18x12-inch rectangle. Spread Coconut Filling over dough, leaving a ½-inch border on one long side. Starting with opposite long side, roll dough into a tight log. Trim ends, and cut into 12 rounds. Transfer rounds, cut side up, to prepared pan, and cover with plastic wrap. Refrigerate for at least 8 hours or overnight (or let rolls rise at room temperature for 1½ hours).
Preheat oven to 375°F (190°C).
Uncover rolls, and bake until puffed and golden brown throughout, about 35 minutes. Let cool for 10 minutes. Drizzle with Coconut Icing while still warm, and sprinkle with toasted coconut flakes before serving.
Recipe by Bake from Scratch at https://staging.bakefromscratch.com/coconut-buns/