In a stand mixer with paddle attachment, mix flour, baking soda, salt, cinnamon, brown sugar, white sugar, eggs, vanilla, and applesauce until batter is formed.
Remove bowl from mixer and gently mix sour cream in by hand with a rubber spatula.
Fold in apples.
Fill muffin cups about 2/3 full.
Bake in 350 degree oven for about 25 minutes. Muffins are done when a toothpick inserted in the middle comes out clean.
When muffins are done, remove from pans and allow to cool on cooling rack.