In the bowl of a stand mixer, combine yeast, 1 teaspoon sugar, and warm milk. Let stand for 5 to 7 minutes until mixture is foamy.
Add remaining sugar, egg yolks, and egg. Whisk together until well incorporated.
In a large bowl, whisk together flour and salt. Add half of flour mixture to mixing bowl. Attach dough hook to stand mixer and mix on low speed until dough resembles a shaggy mess, about 2 minutes. With the mixer running, add softened butter one tablespoon at a time. Add remaining flour mixture and continue to knead at medium speed until dough is smooth and no longer sticks to the sides of the mixing bowl, about 4 minutes.
Transfer dough to a greased large bowl. Cover with plastic wrap and let dough rest at room temperature for 70 to 90 minutes, until dough has doubled in volume.
Punch down risen dough. Recover with plastic wrap and let chill in the fridge for at least 30 minutes (up to 4 hours), until dough is cold.
Place stemmed figs and water in a small heavy-bottomed saucepot. Set over medium heat and bring to a boil. Reduce to a simmer and allow figs to soften, about 10 to 15 minutes. Drain figs and discard water.
In the bowl of a food processor, combine warm figs, chocolate, cocoa powder, powdered sugar, butter, vanilla, and salt. Process until mixture resembles jam. Scrape down bowl as necessary to ensure thorough mixing. Allow mixture to cool slightly before adding egg whites. Pulse 2 to 3 times until egg whites are just incorporated. Transfer to an airtight container and keep chilled in the fridge until assembly.
Working on a lightly floured surface, roll chilled dough to an 18x14-inch rectangle. Reserve about 2 tablespoons of chocolate fig filling. Spread remaining filling over dough, leaving a ½-inch border around the edges, so that it doesn’t ooze out. Beginning at the longer end, roll dough into a tight cylinder. Pinch the seams closed.
With dough seam side up, spread the reserved filling over the seam. Bring one end of dough over to the other end and then pinch the ends together, thus sandwiching the filling, to create a log. You’ll see some filling on the outside.
Gently twist the log, left hand going one way and right hand going the other, until it resembles a figure eight with swirls of filling peeking through. Place twisted roll in a greased 9x4-inch loaf pan. Gently press dough to distribute it to the edges of the pan. Loosely cover with plastic wrap. Let dough rest at room temperature until dough is level with the lip of the loaf pan, about 70 minutes.
Whisk together egg, water, and salt until well combined.
Preheat the oven to 350°F. Lightly brush the top of the risen dough with egg wash. Bake for 45-50 minutes, rotating loaf midway through baking. Bake until the top of the bread is golden brown and the internal temperature of bread registers at least 190°F. Allow bread to cool in pan for 15 minutes. Remove bread from pan and allow it to cool to room temperature on a wire rack, about 1 hour. Allow bread to completely cool before slicing with a serrated knife.