In a medium sized bowl, dissolve the yeast with the warm water. Add the sugar and one of the eggs and whisk to combine. Mix in 1 1/2 cups of flour until the batter is smooth and sticky.
Cover with a towel and let it rest for 30 minutes.
After 30 minutes add the oil and salt. Mix well with a spatula or dough whisk. Add the coconut and remaining 1 cup of flour to the batter and mix until the dough starts to come together.
Transfer dough onto a clean work surface which has been dusted with flour. Knead dough until smooth and elastic.
Place dough in lightly oiled bowl and turn to grease top. Cover; let rise until indentation remains after poking dough with finger down to second knuckle, about 30-40 minutes.See note below for overnight rising.
Punch down dough and transfer to clean surface. Divide the dough into 12 even-sized pieces.
Roll the dough pieces into smooth buns and place close together on a greased 9x13-inch baking tray in a 3-by-4 pattern. Cover with the towel and let rise for about 20 minutes.
Preheat the oven to 375°F.
After the rolls have risen, gently glaze them with the remaining egg (lightly beaten), using a pastry brush. Bake for 20 minutes, or until golden brown. Cool in pan on wire rack.
Bake for 20 minutes, or until golden brown. Cool in pan on wire rack.
Make the icing: Add the powdered sugar to a bowl and mix in the water using a whisk. The icing will seem thick but keep whisking until it becomes smooth and lump free.
Once the buns are cool, dip each one into the icing and letting the excess pour off. Transfer back to the baking pan until every bun has been iced.