1. Place the warm milk and yeast in a large mixing bowl. Let sit for 10 minutes until foamy.
2. Add the sugar, butter, egg, and egg yolk, and whisk to combine.
3. Add the flour and knead with an electric mixer for 8 minutes until you have a soft dough ball. If still quite sticky, add more flour about 2 tablespoons at a time.
4. Transfer to a greased bowl, cover with a towel, and let rise in a warm spot until doubled, about 1 1/2 hours.
5. Once doubled, turn dough onto a well-floured surface and roll into a rectangle approximately 10×18 inches.
6. Spread 3 tablespoons of the softened butter over the rectangle, then sprinkle with 2 tablespoons of brown sugar.
7. In a small skillet, melt the remaining 2 tablespoons of butter. Add the shredded apple, cinnamon, and nutmeg and sauté until the apple is softened, about 5 minutes. Sprinkle the remaining 2 tablespoons of brown sugar over the apple and stir briefly. Spread the apple mixture over the dough.
8. Roll the dough tightly into one long log. Cut in half, then into quarters. Cut each quarter into 3 even rolls — 12 rolls total.
9. Place the rolls in a greased 9×13-inch baking dish. Cover with a towel and let rise at room temperature for 30–45 minutes, until puffed.
10. Preheat the oven to 350°F. Bake for 25–30 minutes, until golden on top.

For the caramel cream cheese frosting:

1. Beat the softened cream cheese and butter together until smooth and fluffy, about 2 minutes.
2. Add the powdered sugar, caramel sauce, vanilla, and salt. Beat on low until incorporated, then increase to medium-high and beat until light and spreadable.
3. If the frosting is too thick, add milk a teaspoon at a time. Spread generously over the warm rolls and serve.