In a heavy skillet set over medium heat, melt the butter and brown sugar until bubbly. Toss in the apple slices and sprinkle with the cinnamon. Cook, stirring frequently, for about 4 minutes. Remove from the skillet to cool slightly. Remove and reserve 24 slices of the apples from the caramel sauce and chop the rest into smaller pieces, stirring them back into the sauce.
Preheat the oven to 375 F and set a rack in the center. Prepare a muffin tin with melted butter, oil or non-stick spray.
In a medium size bowl, whisk together the barley and unbleached flours, brown sugar, cardamom, nutmeg, baking powder, baking soda and salt. In a small bowl, whisk together the vegetable oil, yogurt and beaten eggs. Make a well in the center of the dry mix and add the liquids, whisking briefly to combine. Stir in the chopped apple mixture.
Divide the batter evenly among the 12 muffin cups and top each with 2 of the reserved apple slices. Bake for 22 to 25 minutes or just until the muffins spring back when pressed gently with your finger.
Cool and remove from the pan.