Place the warm milk in a large mixing bowl and sprinkle the yeast over the top. Let sit for 5 minutes for the yeast to activate. Butter a 9-inch square pan and set aside.
Add the eggs, flour, sugar, and salt to the yeast mixture and mix to form a rough dough. Begin piecing in the soft butter and continue to mix and knead until a smooth and elastic dough forms, about 10 minutes.
Place dough in a lightly oiled bowl, cover, and let sit in a warm place until doubled in size, about 90 minutes. Meanwhile, make the fillings.
Melt the butter in a large sauté pan. Add the chopped apples, sugars, and cinnamon and cook until the apples are soft and have released their juices, 5 – 7 minutes. Stir in the lemon juice. Remove the apples from their cooking liquid and set aside. Add 1/4 cup butter to the warm cooking liquid and stir until butter has melted. Set aside.
In a small bowl, mix together brown sugar, cinnamon, and salt. Set aside.
Once dough has doubled in size, transfer to a well-floured surface and roll into a 14x18-inch rectangle. Spread the surface with the 1/2-cup soft butter, leaving a 1/2-inch wide strip empty along the 14-inch top edge. Sprinkle evenly with the cinnamon/brown sugar mixture then top with an even layer of the cooked apples.
Beginning at the bottom of the rectangle, begin to roll up the dough into a long cylinder, sealing along the empty top edge. Cut the cylinder into 12 equal pieces and place on their sides in the prepared 9-inch square pan. Cover and let sit somewhere warm until puffy and nearly doubled in size, about 1 hour. Preheat oven to 350°F.
Pour the reserved apple liquid over the top of the rolls and bake for 35 – 40 minutes. Let cool at least 15 minutes. Meanwhile, make the icing.
Mix together cream cheese, powdered sugar, and vanilla extract until smooth, adjusting consistency with milk. Spread icing over the top of the rolls, the separate and serve.