Add the beaten egg and stir together with a fork or whisk.
1 beaten egg
Pour the wet mixture into the dry ingredients. Stir to combine, then knead for 8–10 minutes, until smooth and elastic. You may need to add a little more flour. If using a stand mixer, the dough should pull away from the sides of the bowl and cling to the dough hook.
Place the dough in a lightly greased bowl, cover, and let it rest for about 10 minutes.
Meanwhile, preheat the oven to 140°F / 60°C for a few minutes, then turn it off. Place the covered bowl inside the warm (but turned-off) oven and let it rise until doubled in size, about 1–1½ hours.
Make the Apple Pie Filling
Peel, core, and finely dice the apples.
In a skillet, melt the butter over medium heat.
2 ½ tablespoons unsalted butter
Add the diced apples, sugar, cinnamon, lemon juice, vanilla, and cornstarch, and stir well.
⅓ cup granulated sugar, 1 ¼ teaspoons ground cinnamon, 2 ½ teaspoons lemon juice, 1 teaspoon vanilla extract, 2 ½ teaspoons cornstarch
Add the water and cook for 5–7 minutes, until the apples soften and the mixture thickens slightly.
3 tablespoons water
Remove from heat and let it cool before using. (I like to spread it on a large plate to speed it up.)
Make the Cinnamon Sugar Filling
In a small bowl, combine the softened butter, brown sugar, and cinnamon until smooth and well blended.
¼ cup unsalted butter, very soft (not melted), ½ cup light brown sugar, 1 ½ teaspoons ground cinnamon
Roll, Fill, Slice & Bake
Roll out the dough on a lightly floured surface into a rectangle (about 14x18 inches / 36x46 cm).
Spread the cinnamon-sugar filling evenly over the surface, then spread the cooled apple filling on top.
Use a pizza cutter to cut 12 even strips.
Starting from the bottom, roll up each strip into rolls and arrange them in a greased 9x13-inch (23x33 cm) baking pan.
Cover with plastic wrap and let rise again until puffy (about 30–45 minutes).
Preheat the oven to 350°F / 180°C.
Remove the plastic wrap, warm the cream slightly, pour it over the rolls, and bake for 25–30 minutes, or until golden on top.
⅓ cup heavy cream
Make the caramel sauce
Heat the cream (don’t boil) and keep it by the stove with the other ingredients.
1 cup heavy whipping cream
Place the sugar in a saucepan over medium heat, but don't stir it right away. After about 3 minutes, it will start melting around he edges. Gently whisk or swirl the pan until all the sugar has melted.
When it begins to foam and small bubbles form, slowly add about one-third of the cream while whisking constantly. Don't add it all at once - it will bubble and steam. Keep whisking, then gradually add the remaining cream in small additions.
Remove the saucepan from the heat, whisk in the butter until smooth, then stir in the salt (if using). Let it cool for a few minutes, then taste and add more salt if you prefer it saltier.
4 tablespoons unsalted butter, cubed, Optional for salted caramel: ½ teaspoon salt or more to taste
Let the caramel cool in a bowl for a few minutes before transferring it to clean, sterilized jars.
Drizzle some caramel sauce over the cinnamon rolls before serving.