Warm the coconut milk to proper bread making temperature. (I test on the inside of my wrist as if warming for a baby - no thermometer required - not too hot). Whisk in the yeast and sugar and set aside for 10-15 minutes and let it bloom (When ready it will appear as a bit of foam or thick material on the surface ). If it doesn't bloom your yeast is bad or your liquid to cold or too hot.
In another bowl combine (dry whisk) the flour, Vital Wheat Gluten, and salt. Then set aside.
Whisk the room temperature egg with the melted but cooled butter (If the butter is too hot it will cook the egg as added). Once the egg and butter are combined, whisk in the Coconut Milk, sugar, yeast mixture.
At this point the Flour mixture can be added to the liquid mixture using any traditional bread making techniques, or continued using a Bread Machine custom set to "Dough" only. Whatever method is used it must satisfy mixing, kneading, and the first rise of the dough.
Punch the dough down and turn out on a floured board. Cut in 12 equal sized pieces for 12 Hamburger buns.
You can form Hamburger Buns with rounds, and Hot Dog buns with a longer shape.
Form your buns, coat with oil or spray oil, and lay out almost touching each other.
I personally use a greased 4 inch deep stainless steel Steamer tray for this. The high sides help make sides on the buns where not touching other buns.
The 4 inch deep tray is placed in a 6 inch deep stainless steel Steamer tray that contains a couple inches of hot tap water. A stainless steal Steamer tray lids goes on top. This is placed on the warm stove, This becomes the 2nd rise proofing chamber. But any successful method you use to allow the dough to rise will work.
Let this 2nd rise go 45-60 minutes or until your buns look doubled in size.
Preheat oven to 350°F. Place the buns in the oven and bake for at least 9 minutes. Take a peek at 9 minutes without letting all the heat escape. They should just be beginning to brown. Often they will not at this point. Check every 2 minutes until they just begin to brown. Then remove from the oven. Remove from the pan and place on a wire rack to begin cooling. DO NOT OVER BAKE or the buns will not be soft.
While still very warn use a piece of buttered wax paper (old butter wrap works great) and lightly butter all surfaces of the bun to create a soft surface. Once cooled, bag and twist tie to avoid having the buns dry out.