Chocolate Babka (Bread Machine Dough + Oven-Baked Finish)

Ingredients

• ½ cup half & half (or ¼ cup milk + ¼ cup heavy cream)
• 2 tablespoons granulated sugar
• ¾ teaspoon salt
• 2 large eggs
• 3 tablespoons unsalted butter (chopped)
• 2½ cups + 1 tablespoon all-purpose flour
• ¾ teaspoon instant yeast
• ⅓ cup unsalted butter
• 2.5 ounces bittersweet or semi-sweet chocolate
• ⅓ cup + 1 tablespoon powdered sugar
• 3 tablespoons Dutch-process unsweetened cocoa
• ⅛ teaspoon cinnamon (optional)
• 1 large egg
• 1 tablespoon water
• pinch table salt
• ¼ cup granulated sugar
• ¼ cup water