Place in the oven and bake for 45 minutes until very soft.
Allow the yam to cool slightly and then peel the skin off.
Place in a saucepot with the other paste ingredients and mash with a potato masher.
Turn heat on medium-low and cook while stirring constantly for 15 minutes.
Transfer mixture to a bowl and allow to cool at room temperature.
Place in the refrigerator to chill.
Babka Dough
Add the yeast, coconut milk and sugar to the bowl of a stand mixer, mix together and let sit for 5 minutes.
Add the butter, egg and salt and mix, using the dough hook, on medium-low speed.
Slowly add the flour and mix until well incorporated. Continue kneading the dough in the mixer for 5 minutes. Dough should be quite sticky.
Form dough into a ball and place in a greased bowl. Cover and allow to rise for 2 hours.
Grease a 9-inch loaf pan and line with a piece of parchment paper, long enough to hang over the edges.
Flour a work surface and roll the dough into a 10 by 18 inch rectangle.
Spread the ube filling over the dough.
Sprinkle the toasted coconut on top and then starting with a long side roll the dough into a cylinder.
Cut the cylinder in half lengthwise, and with the cut sides up wrap each piece over the other, like a rope. Place the wrapped dough into the loaf pan and allow to rise for 1 hour.
Preheat oven to 350 degrees. Brush the bread with egg wash and place in the oven to bake for 50 to 60 minutes.
Topping
Bring the sugar and water to a boil and simmer for 5 minutes. Set aside to cool.
When the bread is done baking remove from oven and brush with syrup, then sprinkle with toasted coconut shavings.
