Whisk together flour, ⅓ cup of sugar and salt in a bowl. Warm milk in the microwave for about 45 seconds or until a thermometer reads 100 to 110 degrees. Add the milk and 1 teaspoon sugar to the bowl of a stand mixer fit with a dough hook. Sprinkle the yeast over the milk and stir to combine. Let stand for about 5 minutes until the yeast starts to foam.
Add eggs, vanilla, flour, sugar and salt to the yeast mixture. Mix on low speed until combined and then increase speed to medium low and mix for about 5 minutes until the dough is smooth.
Add the softened butter by the tablespoon and mix on medium low about 4 minutes until butter is incorporated and dough comes together but is sticky. Scrape down the sides of the bowl as needed during mixing.
Cover the bowl with plastic wrap and let rise in a warm place for 1 to 1 ½ hours until the dough is about doubled (or let rise in the refrigerator overnight). Punch down the dough and scrape out onto your surface. Divide the dough into 2 halves and shape into a rectangle. Wrap each disc in plastic wrap and left refrigerate for 30 minutes to firm up the dough before rolling it out.
Once chilled slightly, starting with one disc, roll the dough out on a floured surface to a 12 x 16 rectangle. Use a knife to lightly score the dough in half horizontally and then again on each half horizontally. (Do not cut through the dough! This is just a visual to spread each color of cream cheese inside the lines). You’ll have 4 long rectangles. Spread one color of cream cheese mixture in each of the 4 rectangles leaving a small ¼ inch border on the top edge of the top color. See the pictures above!
Starting with the long side without the border, roll the dough into a tight log.
Trim off about an inch from each side. Slice the dough down the middle lengthwise into 2 long halves with the layers exposed. Place the end of one of the halves over the top of the other half, pressing together lightly (see pictures above) and then braid the 2 pieces over one another to the bottom, again pressing together lightly. Repeat the last 2 steps with the other dough disc.
Line two, 9 x 5 loaf pans with parchment paper leaving an overhang on the long sides. Carefully place the braided dough into each loaf pan, squeezing the ends slightly to fit if needed. Cover with plastic wrap and let rise in a warm place for an hour.
Bake at 350 degrees for about 30-40 minutes or until golden brown and a tester comes out clean. You can also test it with an instant read thermometer. It should be about 185-190 degrees when done. If the top is browning too much, place a sheet of foil over it to allow it to cook through without the top burning. Immediately after taking out of the oven, brush each loaf with half the sugar syrup, using it all.
Cool a few minutes in the loaf pan and then transfer the babka using the parchment paper to a wire rack to cool completely.