This is my family's favorite breakfast bread. The bread machine does the work, and the filling makes it a treat. Be careful when chopping the walnuts, as if they are too finely chopped, the filling gets mushy. Enjoy!
Place bread flour, 3/4 cup milk, 1/4 cup unsalted butter, 1/4 cup white sugar, 1 egg, 1 tablespoon cinnamon, yeast, and salt into the pan of your bread machine in the order recommended by the manufacturer. Select the Dough cycle and press Start. When the cycle is complete, remove the dough, punch down, cover, and let rest for 10 minutes.
Preheat an oven to 275 degrees F (135 degrees C). Spread the walnuts onto a baking sheet, and toast until the nuts start to turn golden brown and become fragrant, about 45 minutes. Watch the nuts carefully as they bake, they burn quickly. Once toasted, set the nuts aside to cool to room temperature.
Stir walnuts, brown sugar, 3 tablespoons white sugar, and 1 tablespoon cinnamon together in a bowl. Add 4 teaspoons milk and 1 tablespoon butter; mix well.
Turn dough out onto a lightly floured surface and roll into a 15x12-inch rectangle. Cut into three 15x4-inch strips. Spread 1/3 the walnut mixture down the center of each strip. Moisten and roll the long edges of each strip together over the filling, and pinch to seal each rope.
Line up the 3 ropes 1 inch apart on an ungreased baking sheet. To ensure an even braid, start braiding in the middle. Bring the left rope under the center rope, pass the right rope under the new center rope, and repeat braiding to the end. Starting in the middle again, pass the outside ropes over the center rope, repeating to the end. Press rope ends together and tuck in on each side. Cover loaf with a damp cloth and let rise until doubled in volume, about 45 minutes.
Preheat oven to 350 degrees F (175 degrees C).
Beat water and egg together in a small bowl. Stir 2 tablespoons white sugar and 2 teaspoons cinnamon together in a separate bowl. Brush loaf with egg mixture and sprinkle with cinnamon sugar.
Bake in preheated oven until the top is golden brown and the bottom of loaf sounds hollow when tapped, 20 to 25 minutes.