Cook the apples: Melt 1 tablespoon butter in a skillet over medium heat. Add apples, 2 tablespoons brown sugar, 1 teaspoon cinnamon, and lemon juice. Cook 6 to 10 minutes until just tender and most loose liquid has evaporated. Mix cornstarch with cold water, stir into the apples, and cook 1 to 2 minutes until thickened. Spread on a plate and cool completely.
Prepare the yeast: For active dry yeast, stir yeast into warm milk with 1 teaspoon of the measured granulated sugar. Let stand 5 to 10 minutes until bubbly. For instant yeast, mix yeast with the starting flour instead.
Mix the dough: Combine milk, yeast, granulated sugar, melted butter, egg, salt, and 480 g flour. For active dry yeast, add only the remaining granulated sugar.
Knead: Knead with a dough hook for 6 to 8 minutes or by hand for 8 to 10 minutes, until smooth and elastic. Add extra flour one tablespoon at a time only as needed. The dough should remain soft and slightly tacky.
First rise: Place in a lightly greased bowl, cover, and let rise until roughly doubled, about 60 to 90 minutes.
Prepare the pan and filling: Grease a 9-by-13-inch pan. Mix ½ cup brown sugar, 1 tablespoon cinnamon, and 1 teaspoon vanilla.
Fill: Roll dough into an 18-by-12-inch rectangle. Spread with ¼ cup softened butter, leaving a ½-inch strip bare along the far long edge. Add the cinnamon sugar mixture and distribute the cooled apples evenly.
Cut: cut about a 1-inch thick strips with a pizza cutter edge to edge then roll from one edge to the next. No floss needed!
Second rise: Cover and let rise 35 to 60 minutes until puffy. Preheat the oven to 350°F toward the end of the rise.
Bake: Bake 28 to 35 minutes, loosely covering with foil if needed. Check that the middle rolls are baked through, with the bread portion around 190°F. Cool 15 to 20 minutes.
For the simple powdered sugar glaze: Whisk powdered sugar, 1 tablespoon milk, and vanilla until smooth. Add more milk a teaspoon at a time until the glaze is thick but pourable. Spoon over rolls that have cooled for 15 to 20 minutes, then drizzle with ¼ cup caramel sauce.For a thicker coating, use less milk and let the rolls cool until barely warm before glazing. Make frosting, if using: Beat cream cheese and 2 tablespoons softened butter until smooth. Beat in powdered sugar, ¼ cup caramel, vanilla, and salt.
Finish: Spread frosting or glaze over warm rolls. Drizzle with the remaining ¼ cup caramel and serve.