In a small pot, mix the flour and water over medium heat until it becomes a pudding texture. When finished, transfer into a bowl to let it cool.
In a mixing bowl, put together the flour, sugar, salt, and yeast and mix.
In the dry ingredients, mix in the warm milk and egg. You can use a wooden spoon or just your hands. It will form a shaggy dough.
Once the flour and water pudding is cooled down, mix that into the shaggy dough. Once that is thoroughly mixed in, mix in the softened butter.
Once the dough is all mixed together, prepare a floured work surface to knead. The dough will be really sticky and wet at first but as you knead and incorporate more flour, the dough will become firm and less sticky. Knead for about 15 minutes until firm dough is made but still soft and has a sticky touch to it.
Set the dough in a large bowl and cover with plastic wrap or a slightly damp towel. Place it in a warm area and let it rise until doubled in size. The oven with the lamp on is good enough of a warm space.
Once the dough has doubled in size, remove it from the bowl and separate into 12 pieces by first cutting the dough in quarters and then cutting the quarters into threes. Roll each piece into a circle and place of a baking sheet with parchment paper. 6 on a sheet, 2 sheets.
Place the tray back in a warm area to let it rise again. The dough may even rise to double its size again.
Once the dough is risen again, remove from the warm area. Set the oven to 350F.
Take the remaining egg and mix in a splash of milk. Brush this mixture on the top of each bun and bake the buns for 15-20 minutes.
