Chocolate Babka

Name: Chocolate Babka
Prep time: 3 h
Cook time: 50 min
Total time: 5 h
Yield: 10
Ingredients: Ingredients; 3 1/2 cups all-purpose flour; 1/3 cup granulated sugar; 2 1/4 teaspoons active dry yeast (1 packet); 1 teaspoon salt; 1/2 cup whole milk, warmed; 3 large eggs, room temperature; 6 tablespoons unsalted butter, softened; 1 teaspoon vanilla extract; 1/3 cup water; 1/3 cup granulated sugar; 1 egg beaten with 1 tablespoon water (egg wash); 2 tablespoons pearl sugar or coarse sugar (optional); Chocolate Filling; 6 oz dark chocolate (60-70%), finely chopped; 1/2 cup unsalted butter, softened; 1/2 cup powdered sugar; 1/4 cup unsweetened cocoa powder; 1/4 teaspoon salt; 1 teaspoon vanilla extract