Heat butter, sugar, water and milk to 120F.Combine 2 cups flour, salt and yeast in a medium bowl. Make a well in the center. Add butter mixture. Mix in additional ½ cup flour and egg. Knead on low speed for about 7-8 minutes scraping down bowl as needed. Add additional ½ 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 in bulk.Meanwhile, over low heat combine apples, butter, sugar, cinnamon and cornstarch. Heat until apples are softened, about 5-7 minutes. Set aside to cool.In a small bowl combine cream cheese, sugar, vanilla and lemon juice. Mix until smooth.After dough has doubled in bulk. On floured parchment or silicone mat roll out dough into a large rectangle about 12 x 10.Spread cream cheese filling down the center. Arrange apple mixture on top.Using a pizza cutter cut diagonally from end of dough towards center in 1 inch strips. Do this on 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.Cover loosely with plastic. Let rise for about 30-40 minutes.Bake at 350 for about 20-25 minutes or until braid is golden brown.Combine powdered sugar, milk and vanilla. Drizzle over braid. Serve warm or allow to cool and then serve. Delicious either way.