In a stand mixer with the dough hook attachment, add instant yeast, cinnamon, nutmeg, apple cider, flour, vanilla extract, salt, butter, and brown sugar. Mix for 8-10 minutes, starting on low and gradually increasing to medium high speed when dough begins to form.
Add dough to a well oiled bowl and cover with a towel or plastic wrap. Allow dough to rise for an hour in a warm place.
While dough is rising, chop apples and add to a sauce pan. Also add apples, butter, brown sugar, cinnamon, apple cider, and a pinch of salt. Bring to a boil and boil for 5 minutes. Reduce heat to medium low and add ¼ cup water and simmer until apples are soft. Set aside and allow to cool.
Add dough onto a floured work surface and roll into a 12 by 18 inch rectangle. Add apple filling to dough and spread evenly across. Roll dough into the log and cut into 1.5 inch pieces and add to a baking pan (9 x 13) or a cast iron pan (10 inch). Cover with a towel and let rise for 30 minutes.
Preheat oven to 350 degrees F. Bake for 22-25 minutes, until golden brown. Allow to cool.
To make the browned butter frosting, brown butter in a skillet and add to a stand mixer and add powdered sugar, cream cheese, pinch of salt, and vanilla extract. Whip until all ingredients are combined. Slater icing on top of cinnamon buns!