In the bowl of a stand mixer fitted with the dough hook attachment, combine flour, sugar, and yeast. With mixer on low speed, add eggs, milk, and vanilla. Beat until mixture comes together, 2 to 3 minutes. (If the mixture remains too dry and crumbly, add more milk, 1 tablespoon at a time.)
With mixer on medium speed, add butter, 1 tablespoon (14 grams) at a time, letting each piece incorporate before adding the next. Add salt, beating just until combined, about 3 minutes. Increase mixer speed to medium, and beat until a smooth and elastic dough forms and pulls away from the sides of the bowl. (If dough does not pull away from the bowl, add more flour, 1 tablespoon [8 grams] at a time.)
Spray a large bowl with cooking spray. Place dough in bowl, turning to grease top. Cover and let rise in a warm, draft-free place (75°F) until doubled in size, 1½ to 2½ hours. After dough has risen, refrigerate for 30 minutes. Alternatively, the dough can be made one day in advance and the entire rise may take place in the refrigerator overnight.
For egg wash: In a small bowl, lightly beat together egg and water.
Spray 2 (9x5-inch) loaf pans with cooking spray, and line with parchment paper.
Divide dough in half. On a heavily floured surface, roll half of dough into a 16x12-inch rectangle. Brush edges of dough with egg wash. Spread with half of Chocolate Filling, leaving a 1-inch border on long sides of dough. Starting at one short side, roll up dough, jelly-roll style, and press edge to seal. Using a sharp knife, cut roll in half lengthwise. Carefully twist dough pieces around each other, and place in prepared pan, cut sides up. Repeat procedure with remaining half of dough. Place prepared pans on a foil-lined rimmed baking sheet. Cover and let rise in a warm, draft-free place (75°F) until doubled in size, 1 to 1½ hours.
Preheat oven to 350°F. Bake for 30 minutes. Cover with foil, and bake until a thermometer in center registers 190°F, about 35 minutes more.
For simple syrup: In a small saucepan, bring sugar and ½ cup water to a boil over medium-high heat, stirring occasionally, until sugar is dissolved. Remove from heat, and let cool slightly. Pour simple syrup over warm loaves while still in pans. Let cool in pans for 5 minutes before transferring to a wire rack.