In a large mixing bowl, combine 2 and 1/4 cup all-purpose flour, sugar, salt and active dry yeast. Whisk well.
In a microwave-safe bowl, microwave butter, milk and water until the butter melts. The temperature of the mixture should be between 110-115 F. Re-heat in the microwave when necessary.
Add the butter mixture to the dry ingredients in the mixing bowl. Stir the mixture using a wooden spoon until incorporated. Add the egg and stir. Add some of the reserved flour gradually while stirring until the mixture turns into a soft dough that gathers in the center of the bowl. You may not need all the flour.
Turn the dough onto a floured board and knead for 6-12 minutes, until it is smooth and elastic. Let the dough rest for ten minutes in a bowl, covered.
Roll the dough using a rolling pin into a 14x10 inch rectangle. Spread the ube across the dough. Roll the short side of the dough to form a tight log. Cut the log into 4 equal portions using a sharp knife. Arrange the portions, cut side up in a greased 9x5 loaf pan.
To let the rolls rise, preheat oven to 200 F. Once the temperature is reached, turn OFF the oven and put the ube loaf inside. Let it rise for 60-90 minutes, or until the size is doubled. Take the ube loaf out of the oven. Preheat oven to 350 F. Brush the surface with egg wash. Bake the loaf for 18-22 minutes or until the top is golden.