In a stand mixer bowl, combine flour, sugar, yeast, and salt. Add eggs, milk, and vanilla. Mix on low to combine.

Knead on medium-low for 5-7 minutes until dough is smooth. Slowly add softened butter, one tablespoon at a time, kneading until fully incorporated and dough is soft and shiny (about 10-15 minutes).

Place dough in an oiled bowl, cover, and let rise in a warm place for 1.5-2 hours until doubled. (Alternatively, rise for 45 mins then refrigerate overnight).

Make the filling: Melt chocolate and butter together. Whisk in powdered sugar and cocoa powder until smooth. Let cool slightly, then stir in finely chopped pistachios.

Grease and line a 9x5-inch loaf pan with parchment paper.

Roll the risen dough into a 16x12-inch rectangle on a floured surface. Spread the filling evenly over the dough, leaving a small border on one long edge.

Roll up tightly into a log starting from the long edge with filling. Pinch the seam to seal. Place seam-side down.

Using a serrated knife, cut the log in half lengthwise. Twist the two halves together, keeping the cut sides facing up. Pinch ends to seal.

Place the twisted loaf into the prepared pan. Cover loosely and let rise for 45-60 minutes until puffy.

Preheat oven to 350°F (175°C). Bake for 40-50 minutes until deep golden brown (internal temp approx. 190°F/88°C). Tent with foil if browning too fast.

While baking, make syrup: Simmer sugar and water in a saucepan until sugar dissolves.

Brush the hot syrup over the babka immediately upon removing from the oven. Let cool in pan for 15 minutes, then transfer to a wire rack.

Once cool, drizzle with melted chocolate and sprinkle with roughly chopped pistachios.