Make the tangzhong. Heat water with milk and bread flour in a small pot over low heat. Whisk the mixture constantly until it turns into a thick paste, about 3-4 minutes. Remove from heat and cool completely.
Combine the tangzhong with bread flour, warm milk, granulated sugar, instant yeast, softened butter, egg, egg yolk, and salt. Knead the dough until smooth, about 7-8 minutes. Transfer to a clean bowl, cover, and proof in a warm area until doubled in size, about 1 hour.
Meanwhile, make the sun-dried tomato pesto. In a food processor or blender, combine toasted walnuts, grated parmesan, basil, sun-dried tomatoes, and garlic. Pulse until roughly chopped. With the machine running, slowly add olive oil and jarred sun-dried tomato oil until well combined. Season with salt and pepper and set aside.
Preheat oven to 350 degrees F. Grease a 9-inch springform pan and place a 4-inch oven-safe ramekin in the center.
Punch down dough. Lightly dust a clean work counter with flour and roll out the dough into a 12x18-inch rectangle. Spread about ¾ of the pesto in an even layer on the dough, leaving a ½-inch border. Roll the dough lengthwise into an 18-inch log. Cut the roll in half lengthwise and face the cut sides up. Braid the two ropes together to create a twist.
Transfer the log to the prepared pan, shaping it into a circle. Tuck the ends together to enclose the circle. Lightly cover the bread and proof for 45 minutes in a warm area.
Place the baking pan on a sheet pan to catch any oil that may leak. Bake the babka for 25-30 minutes or until golden brown. Remove from the oven and let cool slightly. Serve warm or at room temperature.