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In a medium bowl, stir together ¼ cup of the brown sugar, and 1 teaspoon of the cinnamon. Stir in the apple cubes and set aside.
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In a large bowl, using a hand mixer, beat together the oil, granulated sugar, and the remaining ¾ cup brown sugar until combined. Add the eggs, sour cream, and vanilla and mix until smooth.
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In a separate bowl, whisk together the flour, baking powder, nutmeg, the remaining 1 teaspoon cinnamon, and the salt. Slowly add the dry ingredients to the wet ingredients and mix until well combined. Fold in the cinnamon apples. Cover and refrigerate for 15 minutes.
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Preheat the oven to 425°F. Line two muffin tins with paper liners or spray with nonstick cooking spray.