In a medium bowl, whisk together brown sugar, flour, and cinnamon. Cut in butter until coarse crumbs form. Set aside.
Preheat oven to 375F. Line muffin tins with paper baking cups or coat with cooking spray.
In a mixing bowl using a handheld or stand mixer, cream together butter and sugars until smooth, about 1 minute. Beat in sour cream, vanilla, and eggs one at a time.
In another bowl, whisk together flour, baking soda, baking powder, cinnamon, nutmeg, and salt. Add dry ingredients to wet ingredients, and mix until just combined (do not overmix). Fold in apples.
Fill muffin tins about ¾ full and sprinkle crumb mixture over muffin batter. Bake for 15 to 20 minutes, or until toothpick inserted in the center comes out clean.