Make the dough: Coat an 8 1/2-inch loaf pan with 1 tablespoon of the unsalted butter. Place the remaining 7 tablespoons unsalted butter in the bowl of a stand mixer fitted with the dough hook and let sit at room temperature until softened.
Fill a liquid measuring cup with 1/4 cup warm water. Add 1 packet active dry yeast and 1 teaspoon of the granulated sugar and stir to combine. Let sit until foamy, about 10 minutes.
Add the yeast mixture, remaining 1/4 cup granulated sugar, 2 cups all-purpose flour, 1 large egg, 1 large egg yolk, 1 teaspoon vanilla extract, and 1/4 teaspoon kosher salt to the stand mixer. Mix on low speed until combined, stopping and scraping down the sides of the bowl a few times as needed. Continue to mix on medium speed until a soft, smooth dough forms, about 2 minutes. If the dough is very sticky, mix in more flour 1 tablespoon at a time (do not add more than 4 tablespoons).
Cover the bowl and let rise in a warm place until increased in volume and sponginess (the dough does not need to double), about 1 hour. Meanwhile, make the filling.
Make the filling: Coarsely chop 1 1/2 ounces bittersweet chocolate (about 1/4 cup). Place in a medium-sized microwave-safe bowl. Cut 2 tablespoons unsalted butter into 2 pieces and add to the bowl. Microwave 1 minute, then whisk until smooth and combined. If the chocolate and butter are still unmelted, microwave for 30 seconds more.
Add 1/4 cup packed light brown sugar, 3 tablespoons light corn syrup, 1/2 teaspoon vanilla extract, and 1/8 teaspoon kosher salt, and whisk well to combine.
Assemble the babka: Arrange a rack in the middle of the oven and heat the oven to 350°F. Coarsely chop 1 cup pecan halves. Reserve 1 tablespoon for sprinkling on top of the babka.
Beat 1 large egg white in a small bowl. Sprinkle a work surface with a little flour. Roll the dough into a large, thin rectangle, about 10 inches tall and 12 inches wide, with a longer side closer to you.
Spread the chocolate filling onto the dough rectangle, leaving a 1/2-inch border at the top. Sprinkle with the chopped pecans. Start from the bottom edge, roll the dough up into a tight log. Pinch the dough together to seal the seam. Using a large serrated knife, cut the log in half lengthwise, creating two halves with lots of layers.
Turn the halves so that the layers are facing up. Press the two halves together at one end, then twist the halves around each other, creating a spiral. Press the halves together again at the other end. Flour your hands and carefully lift the loaf into the loaf pan. Brush the top of the loaf with a thin layer of egg white. Sprinkle with the reserved 1 tablespoon chopped pecans.
Bake until the top is browned and firm to the touch, 40 to 45 minutes. You can also check by inserting a skewer or instant-read thermometer into the center of the loaf — it should come out with a few crumbs or register at least 210ºF. Let cool in the pan on a wire rack for 20 minutes. Remove the babka from the pan and let cool 10 minutes more before slicing.