First, make the dough. In the bowl of a freestanding electric mixer fitted with a dough hook, combine 4 1/4 cups all-purpose flour, 1/2 cup granulated sugar, 2 teaspoons instant yeast, and 1/4 teaspoon kosher salt. Whisk to combine and nestle into the electric mixer. Turn the mixer on to its lowest speed and add 3 large eggs and 1 large egg yolk one at a time, only adding the next egg when the previous egg is fully incorporated. With the mixer still on the lowest setting, stream in 1/2 cup warm water. Once all the water has been added, increase the mixer speed to medium and mix until dough forms, around 3 minutes. Add 2/3 cup cubed unsalted butter a few cubes at a time, adding butter only when the previous cubes have fully incorporated into the dough. Once all the butter has been added, use a rubber spatula to scrape down the sides of the bowl to incorporate any stray ingredients. Turn the mixer back to medium speed and knead for 10 minutes, until the dough is completely smooth, elastic and shiny.
Once the dough is as described, transfer the dough to a lightly greased bowl. Cover with plastic wrap and let rise at room temperature (between 70 to 80 degrees) for at least 2 hours, until the dough has doubled in size.
While the dough is rising, make the ube filling and prepare your pans. In a medium bowl, combine 1/3 cup ube halaya jam, 1/3 cup crème fraîche, 1/3 cup granulated sugar, 2 teaspoons ube extract, and a pinch of kosher salt. Set aside. Prepare a 13 x 4-inch pullman loaf pan by using a 12 x 16-inch piece of parchment paper to line the bottom of the pan and create a "sling" with overhanging edges on the long sides of the pan. In order to achieve this, you'll need to align the center of the parchment paper lengthwise over the pan, and use your hands to press the center of the paper into the bottom edges of the pan. Run a butter knife over the parchment paper across the bottom edges of the pan to get the paper to adhere and mold to the pan's shape. This is important! You'll need this for later. It might even be beneficial to put some heavy objects in the pan (like clean cutlery) to get the parchment paper to keep its shape. Once the dough has doubled in size, turn the dough out onto a lightly floured surface. Use a rolling pin to punch down the dough and roll out to a rectangle about 20 x 13-inches wide. Use a bench scraper to push the edges into the dough mass, creating smooth, even sides for the slab of dough. Use an offset spatula and spread the ube filling (from the 3rd step) evenly across the dough, making sure to spread filling all the way up to the edges of the dough. Once all the filling has been used, it's time to roll and braid the babka. Use both hands to roll the rectangle like a roulade, starting from one of the shorter 13-inch side of the rectangle. You'll end up with a thick log of dough that is 13-inches long and looks and feels a little bit like a massive burrito. Transfer the dough to a large chopping board and use a sharp, serrated knife to slice it in half lengthwise. You are essentially dividing the log into two long even halves, with the layers of dough and filling from the log visible along the length of both halves. Scoot both halves to the side of the chopping board to clear a work space.
Take the parchment paper sling that's lining the pullman loaf pan and place it on the center of the chopping board. Spray the parchment paper with a light layer of cooking spray. Transfer both dough log halves to the center of the parchment paper. If you've done the 3rd step properly, the parchment paper should still retain the shape of the loaf pan — you'll need to put the logs in the portion of the parchment paper that fits to the the bottom of the loaf pan. Align the dough logs so that they are next to each other and parallel, but with the layers of dough and filling visible and facing upwards. Gently press together one end of each half, and then lift the right half over the left half. Repeat this process, but this time, lift the left half over the right to create a simple, two-pronged plait. Repeat this process until there is no longer any dough left to braid — I think I was only able to cross the dough around 3 times or so since it's fairly thick but and short. Squeeze together the ends.
Lightly spray the loaf pan with cooking spray. Use the parchment paper sling to transfer the dough braid into the loaf pan. Cover with plastic wrap and let rise at room temperature (between 70 to 80 degrees) for at least 1 to 1 1/2 hours. During this time, the cake will rise by about 10 to 20 percent.
About 30 minutes into the rising process, prepare your oven and make the syrup. Center a rack and preheat to 375 (F). Combine 1 1/4 cups granulated sugar and 2/3 cup water in a small saucepan over medium heat. Whisk until the sugar is completely dissolved, and bring to a boil. Remove from heat and transfer to a wire rack to cool.
Once the babka has finished its second rising, remove the plastic wrap from the loaf pan and transfer to the preheated oven. Bake for 30 minutes, before removing the babka from the oven and loosely covering the top of the loaf with a sheet of aluminum foil. Return to the oven and bake for an additional 15 to 20 minutes, or until a skewer inserted into the center of the babka comes out clean.
Transfer to a wire rack and immediately use a pastry brush to brush the syrup (from the 8th step) across the babka. It's important to use all the syrup — it will seem like its too much, but I promise that it's not! Allow the babka to cool in its pan for an additional 30 minutes, before using the parchment paper sling to remove it from its pan and onto the wire rack to cool completely.