For the dough, pour the warm milk into the bowl of a stand mixer and sprinkle the yeast over top. Let it sit for 5-10 minutes until foamy. Add the sugar, eggs, flour, and salt. Mix with the dough hook on low speed until a shaggy dough forms. Add the softened butter cubes and increase the speed to medium. Knead for 5-7 minutes, until the dough is smooth and elastic.
Place the dough in a lightly oiled bowl, cover it with a towel, and let it rise in a warm place for about 1 hour, or until doubled in size.
While the dough rises, make the apple filling. In a small saucepan, combine the diced apples, sugar, water, lemon juice, spices, and cornstarch. Cook over medium heat, stirring occasionally, until the mixture bubbles and thickens, about 5-7 minutes. Remove from heat and let it cool completely.
For the cinnamon filling, beat the softened butter, brown sugar, and cinnamon in a small bowl with a hand mixer until light and fluffy.
Once the dough has risen, punch it down and turn it out onto a lightly floured surface. Roll it into an 18x12-inch (45x30 cm) rectangle. Spread the cinnamon filling evenly over the dough, leaving a small border on one long edge. Spread about 1 cup of the cooled apple filling over the cinnamon layer, reserving the rest for topping.
Tightly roll the dough into a log, starting from the long edge opposite the clean border. Pinch the seam to seal. Use a sharp serrated knife to slice the log into 12 equal rolls. Arrange the rolls cut-side up in a greased 9x13-inch (23x33 cm) baking pan.
Cover the pan with a towel and let the rolls rise for another 20-30 minutes. Meanwhile, preheat your oven to 375°F (190°C). Bake the rolls for 25-30 minutes, until golden brown on top.
While the rolls bake, make the icing. Melt the 4 tablespoons of butter in a small skillet over medium heat. Continue to cook, swirling the pan, until the butter foams and brown specks appear at the bottom. Immediately pour into a heatproof bowl to cool. In a medium bowl, beat the softened cream cheese and cooled brown butter until smooth. Add the powdered sugar and vanilla, then beat until combined. Add milk, one tablespoon at a time, until the icing reaches a pourable consistency.
Let the rolls cool in the pan for about 10 minutes, then spread the icing over them. Top each roll with a spoonful of the reserved apple filling before serving.