Place all the dough ingredients in a bowl of a stand mixer. Using a wooden spoon or spatula, mix to combine all of the ingredients.
Place the dough hook in and mix until a smooth and sticky dough forms. About 4-5 minutes.
Place the dough in a large bowl, cover with a towel or plastic wrap and allow to rest until it has doubled.
Cream Cheese Filling:
Place the cream cheese, sugar and ube extract into the bowl of a stand mixer and mix well.
Assembly:
Place the dough on a very lightly floured surface. Divide dough into 2 equal parts. Roll each dough piece into 12" wide and twice as long as the pan you are using.
Spread the cream cheese filling in the center of the dough while leaving a ½ inch around the edges. Spread the ube halaya on top of the cream cheese.
Roll the filled dough starting from the farther edge towards you until you have a long log. Using a serrated knife, gently cut the roll in half lengthwise, starting at the top and finishing at the seam, essentially dividing the log into two long even halves, with the layers of dough and filling visible along the length of both halves. With the cut sides facing up, gently press together one end of each half, then lift the right half over the left half. Repeat this process, but this time lifting the left half over the right, to create a simple two-pronged plait. Gently squeeze together the other ends so that you are left with the two halves, intertwined, showing the filling on top.
Grease two loaf pans. Carefully lift the dough into a loaf pan and let the braided Babkas rise for about 3-4 hours (or overnight).
Simple Syrup:
Combine the water and sugar in a pot at medium heat and let simmer for about 5 minutes. Set aside until babkas are baked.
Final:
Once the babkas have risen, set the oven to 350F. Brush the babkas with the egg wash. Bake the for 30-40 minutes until dark golden and glistening. Take the Babkas out of the oven and brush with the simple syrup.