Apple Cinnamon Rolls with Brown Butter Maple Icing

1. Combine the warm milk, yeast, and 1 tablespoon brown sugar in the bowl of a stand mixer fitted with a dough hook. Leave for 5–10 minutes, until bubbles appear on the surface. 2. Add 4 tablespoons butter, the eggs, 3 1/2 cups flour, and kosher salt. Mix with the dough hook for 4–5 minutes, until the flour is incorporated. If the dough remains sticky, gradually add the remaining 1/2 cup flour, using more if needed, until smooth to the touch. 3. Cover the bowl with plastic wrap and let the dough rise at room temperature for about 1 hour, or until doubled in size. 4. Mix the 1/2 cup brown sugar, granulated sugar, and cinnamon in a small bowl. Grease a 9-by-13-inch baking dish with butter. 5. Punch down the risen dough and roll it on a lightly floured surface into a rectangle approximately 12 by 18 inches. Spread with 6 tablespoons softened butter, then scatter the chopped apples and cinnamon-sugar mixture evenly over it. 6. Starting at a long side, roll the dough tightly into a log and pinch the seam closed. Cut into 12–15 rolls with a sharp knife. Arrange in the prepared dish, cover with plastic wrap, and let rise for 20–30 minutes. 7. Preheat the oven to 350°F. Remove the plastic wrap and bake for 25–30 minutes, until the rolls are golden brown. 8. For the icing, heat the remaining 4 tablespoons butter in a saucepan over medium heat for about 2–3 minutes, until lightly browned and fragrant. Remove from the heat. Whisk the browned butter with the cream cheese, maple syrup, powdered sugar, vanilla, and a pinch of salt until combined. 9. Spread the icing over the warm rolls and serve.