Apple Cinnamon Rolls

Method

1. Prepare the dough: Whisk the warm milk (about 110°F/43°C), 2 Tablespoons of the sugar, and the yeast together in the bowl of a stand mixer fitted with a dough hook. Cover and let sit for 5–10 minutes, until foamy on the surface. (No mixer? Use a large bowl and mix with a wooden spoon or silicone spatula.)
2. Add the remaining sugar (100 g) and the butter and beat on medium speed until the butter is slightly broken up. Add about 125 g (1 cup) of the flour, the eggs, and salt, and beat on low speed for 30 seconds. Scrape down the bowl, then add the remaining flour. Beat on medium speed until the dough comes together and pulls away from the sides of the bowl, about 2 minutes. If it's too sticky, add more flour 1 Tablespoon at a time; if dry and crumbly, add more milk 1 teaspoon at a time.
3. Knead the dough: Beat on low speed for an additional 5–8 minutes, or knead by hand on a lightly floured surface for 5–8 minutes, until smooth, supple and elastic. Add flour only a teaspoon at a time if it becomes too sticky. Poke it—if it slowly bounces back, it's ready to rise.
4. 1st rise: Lightly grease a large bowl with the oil. Place the dough in the bowl, turning to coat. Cover and let rise at room temperature for 1 1/2 to 2 hours, or until doubled in size.
5. Pre-cook the apples: While the dough finishes rising, cook the chopped apples in a skillet with 1 Tablespoon of water over medium heat, stirring frequently, for about 5 minutes, until beginning to soften. Set aside to cool slightly.
6. Grease the bottom and sides of a 9×13-inch (23×33 cm) baking dish or line it with parchment paper.
7. Roll out the dough: Punch down the dough and roll it into a 12×18-inch (30×46 cm) rectangle on a lightly floured surface. If it keeps shrinking back, cover and let it rest for 10 minutes, then continue rolling.
8. Fill the rolls: Spread the softened butter all over the dough. Mix the brown sugar and cinnamon and sprinkle evenly over the butter, then scatter the pre-cooked apples on top. Tightly roll up the dough into an 18-inch (46 cm) log. Cut into 12 even rolls, about 1.5 inches (4 cm) wide, and arrange in the prepared pan.
9. 2nd rise: Cover the pan and let the rolls rise until puffy, about 1 hour.
10. Preheat the oven to 350°F (177°C).
11. Bake for about 26–29 minutes, until lightly browned on top, rotating the pan halfway through. Tent with foil if the tops brown too quickly. Cool the pan on a rack for about 10 minutes while you make the icing.
12. Make the icing: Whisk the powdered sugar, vanilla, milk, and warmed salted caramel together until smooth. Drizzle over the warm rolls and serve. Store leftovers covered at room temperature for up to 2 days or in the refrigerator for up to 5 days.