Prepare Tangzhong: In a medium saucepan, whisk together bread flour and water over medium heat. Cook the mixture for about 3-4 minutes, stirring constantly until it thickens into a smooth paste. Remove it from heat and let it cool completely.

Mix Dough: In a stand mixer, combine the cooled tangzhong with the remaining bread flour, sugar, salt, active dry yeast, whole milk, and egg. Mix on low speed until the dough is combined, then let it rest for 5 minutes. Gradually add in the softened unsalted butter, mixing for an additional 15 minutes until the dough turns smooth and elastic.

Refrigerate Dough: Transfer the dough to a lightly oiled container and cover it securely with plastic wrap. Place it in the refrigerator for 8 to 12 hours overnight.

Make Filling: While the dough chills, prepare the chocolate filling by melting the dark chocolate and butter together in a heatproof bowl over simmering water. Mix in the sifted powdered sugar, cocoa powder, and a pinch of salt until smooth.

Shape Babka: Roll the dough out into a large rectangle on a floured surface. Spread the chocolate filling evenly across the dough and sprinkle with toasted pistachios. Roll the dough tightly into a log, slice it down the center, and twist the two halves together.

Proof: Place the braided babka in a greased loaf pan and cover it loosely with a clean kitchen towel. Let it rise for 60 to 90 minutes or until it has puffed up significantly.

Bake: Preheat your oven to 350°F (180°C). Bake the babka for 30 to 35 minutes or until the top is golden brown and its internal temperature reaches 190°F (88°C).

Syrup Finish: Prepare a simple syrup by simmering equal parts water and sugar in a small saucepan until dissolved. Brush this syrup generously over the warm babka once out of the oven.

Cool: Allow the babka to cool for about 20 minutes in the pan, then transfer to a wire rack to cool completely.