Proof the Yeast. In the bowl of a stand mixer, add the warm water. Sprinkle the yeast and about 1 teaspoon of the sugar. Let this sit until foamy, about 5 minutes.
Add the milk, egg, cream, the remaining sugar and salt. Stir everything with a wooden spoon.
Add 3 cups of flour, one cup at a time, stirring well with the wooden spoon after each addition. Now using the remaining flour, add gradually and stir, just until the mixture gathers into a shaggy ball of dough. You will not need the entire amount of flour here.
Knead the Dough. Attach the dough hook to the mixer. Knead the dough on medium-high speed, gradually adding the remaining flour while the mixer is running. Knead for 5-7 minutes, or until the dough is cleaning the sides of the bowl, and is smooth and stretchy. You may have used the entire amount of flour at this point. If the dough is still sticky and not gathering, sprinkle up to 3 tbsp more of extra flour as you knead. see note 2
Gather the dough into a ball. Place it in a lightly greased bowl and let it rise until doubled, about an hour to an hour and a half.
Shape the Dough. After rising, dust a clean surface with flour. Roll the dough into a rectangle that is about ¼ inch thick. Spread ube jam on the rectangular dough. You will not use all of the ube jam here. Roll the dough into a log starting on the short edge. Then cut the log into 12 equal portions.
Let the rolls rise. Arrange the rolls in a parchment-lined 9x13 inch baking pan. Cover loosely with a towel and let rise for an hour, until they are soft, puffy and touching. Brush egg wash on the surface of the rolls.
Bake the ube buns at 350℉ for 35-40 minutes, or until the tops are golden. Let cool then, spread cream cheese icing on top.