Make the dough: Combine the water, milk, and butter in a small saucepan over medium heat. When butter is completely melted, remove from heat and cool for about 5 minutes (to 120-130°F).
In the bowl of a stand mixer fitted with a dough hook attachment (you may also use a large bowl and mix by hand with a wooden spoon), combine 1 1/2 cups flour with the sugar, cinnamon, salt, and yeast. Add the liquids and mix until combined. Add the egg.
Add the remaining flour 1/4 cup at a time until dough clears the bowl and is still sticky to the touch without coming off on your hands. Continue kneading dough with either the dough hook or by hand until dough is smooth and elastic, about 7-10 minutes.
Put dough in a greased bowl, cover with plastic wrap and place in a warm spot. Let rise until doubled, about 1-2 hours.
Make filling: While the bread is rising, combine the brown sugar, heavy cream, butter, and salt in a saucepan over medium heat. Stir until butter is melted and mixture is smooth. Bring to a boil and let simmer until thickened, stirring frequently, about 5-7 minutes. Remove from heat and add the vanilla.
Let cool to a spreadable consistency. If the caramel becomes too hard, return it to medium low heat to melt it a little before spreading it on the dough.
Prepare the braided loaf: Gently punch down dough to release gases and knead by hand a few times. Let sit for a few minutes.
Roll dough out on a clean surface to about 15x10-inch rectangle. Transfer to a sheet pan lined with parchment paper. Divide dough in thirds lengthwise and cut 1-inch strips crosswise down the length of the outer thirds.
Spread the cooled caramel in the center third. Top with the sliced apples. Fold the cut strips over the filling at a diagonal, alternating side to side.
Brush the loaf with the egg wash. Cover with plastic wrap and let rise while oven preheats to 350°F. Bake until golden brown, about 25-30 minutes. Some of the caramel may ooze out during baking. Let cool on pan for about 5 minutes, then transfer to a cooling rack or serving plate. Dust with powdered sugar.