Lightly butter a 9-inch cake pan, pie pan, or an 8x8 baking pan and line it with parchment paper. Set it aside.
Combine the warm milk, melted butter, sugar, yeast, egg, flour, salt, and vanilla in a stand mixing bowl and mix with the dough hook for 9-10 minutes. The dough should be soft, tacky and just stick to the bottom of the bowl. If it’s sticking to the sides of the bowl, add an extra tbsp of flour.
Transfer the dough to a large lightly buttered bowl and cover tightly with plastic wrap. Let it rise in a warm spot for 30 minutes, until doubled in size. *Place the bowl on top of your oven with a towel underneath and turn it on to 300 degrees F to help it rise quickly!
While the dough is rising, make the apple sauerkraut filling. Add the diced apples, sauerkraut, granulated sugar, 1 tbsp of the butter, and a sprinkle of the cinnamon to a skillet and cook over medium heat for 8-10 minutes until the apples have softened. Set them aside to cool.
When the dough has doubled, lightly flour the top of the dough and your work surface. Press the dough down to release any excess air, then roll it out into a 12x8 rectangle, about ½ inch thick.
Use your hands to spread the softened butter over the dough, leaving a ½ inch edge at the top.
Combine the brown sugar and cinnamon together and sprinkle it over the butter. Add the cooled apple and sauerkraut mixture on top of the sugar and then use your hands to press and stick it into the dough.
Starting at the bottom, tightly roll the cinnamon roll up to the top.
The rolled log should be 12-inches long.
Using a serrated knife, cut about ½ inch off each end.
Then cut 6-2 inch rolls and arrange them in the prepared baking pan with some room around each one.
Cover tightly again with plastic wrap and let rise until almost doubled in size, about 20-25 minutes. While it’s rising, preheat the oven to 350 degrees F. *Place the bowl on top of your oven with a towel underneath and turn it on to 300 degrees F to help it rise quickly.Bake for 20-22 minutes, or until the tops are just golden brown. Let the pan cool slightly on a wire rack while you make the icing.
In a mixing bowl, cream together the cream cheese, butter, and powdered sugar for 2-3 minutes until fluffy and smooth. Add the vanilla extract and mix to combine.
Use a knife or spoon to finish the tops of the cinnamon rolls with the icing and serve warm.
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