1.Heat 6 tablespoons butter, 3 tablespoons sugar, water, and 1/2 cup milk until warm.
2.In a bowl of a standing mixer, combine 2 cups flour, salt and yeast. Make a well in the center. Add warmed butter mixture then mix in 1/2 cup flour and egg.
3.Using a dough hook attachment knead on low speed for 8 minutes scraping down bowl as needed. Add additional 1/2 cup of flour while kneading, if necessary. Dough should be smooth. Place dough in a greased bowl. Cover with plastic wrap. Let it rise for about an hour or until doubled size.
4.Meanwhile, over low heat combine apples, 1 tablespoon butter, 3 tablespoons sugar, cinnamon, and cornstarch. Heat until apples are softened, about 5-7 minutes. Set aside to cool.
5.In a small bowl combine cream cheese, 1/4 cup sugar, 1 teaspoon vanilla, and lemon juice. Mix until smooth.
6.Take the dough out and place on floured work surface. Roll out dough into a large rectangle about 12 x 10. Spread cream cheese filling down the center. Arrange apple mixture on top.
7.Using a pizza cutter cut diagonally from end of dough towards center in 1 inch strips both both sides. Starting on one side of the braid pull one strip up and over the braid. Repeat with the other side. Continue alternating until braid is complete. Place onto a baking sheet lined with parchment paper. Cover loosely with plastic. Let rise for about 30-40 minutes.
8.Preheat oven to 350 degrees. In a small bowl beat 1 egg. Brush egg over the top of the braid and bake for about 20-25 minutes or until braid is golden brown.
9.Combine powdered sugar, 2 3/4 teaspoons milk and 1/8 teaspoon vanilla. Drizzle over braid. Serve warm or allow to cool and then serve.