Whisk the warm milk, 2 Tbsp sugar, and yeast in a mixer bowl. Cover for 5–10 minutes, until foamy. Mix in the remaining sugar and butter. Add 1 cup (125g) flour, eggs, and salt, then add the remaining flour. Mix until soft and tacky; knead for 5–8 minutes, until smooth and elastic.
Grease a large bowl with the oil or butter. Add the dough, cover, and rise for 1½–2 hours, until doubled. Cook the apples with 1 Tbsp water over medium heat for about 5 minutes, until beginning to soften. Cool slightly. Grease or parchment-line a 9×13-inch glass dish.
Roll the dough into a 12×18-inch rectangle. Spread with 6 Tbsp softened butter. Combine the brown sugar and cinnamon and sprinkle over the butter. Add the apples. Roll into an 18-inch log, cut into 12 rolls, and place in the baking dish.
Cover and proof for about 1 hour, until puffy. Preheat the oven to 350°F. Bake for 26–29 minutes, rotating halfway through. Tent with foil if browning quickly. The tops should be lightly golden and the center dough should register approximately 190°F.
Cool for about 10 minutes. Whisk the confectioners’ sugar, vanilla, milk, and warmed salted caramel until smooth, then drizzle over the warm rolls.