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For the cake:
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Preheat oven to 350F.
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Coat an 8-inch-square glass baking dish with cooking spray and dust it with flour, shaking out the excess.
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Spread walnuts on a baking sheet and toast, stirring once halfway, until fragrant, 6 to 8 minutes.
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Transfer to a plate to cool.
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Reduce oven temperature to 325.
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Add whole wheat pastry flour, barley flour, baking powder, cinnamon, cloves, baking soda, nutmeg and salt in a large bowl, whisk until well mixed.
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Add eggs and brown sugar in a medium bowl, whisk until thoroughly blended.
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Mix together yogurt, orange zest and juice in a small bowl and stir until smooth, gradually whisk into the egg mixture along with oil.
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Add the wet ingredients to the dry ingredients in 2 additions, mixing well in between until just incorporated.
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Fold in 1 cup of the walnuts.
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Pour the batter evenly into the prepared baking dish.
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Bake the cake until a wooden stick inserted into the center comes out almost clean, with just a few moist crumbs on, 38 to 46 minutes.
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For the syrup:
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Meanwhile, stir in 13 cup orange juice, 4 tablespoons of brown sugar, orange zest strip and cloves in a small heavy saucepan, bring to a boil over medium-high heat, stirring often.
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Reduce heat to maintain a gently simmer and cook until thickened, about minutes (you will have a scant 13 cup); remove the zest and cloves.
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Allow to cool.
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When the cake is baked, transfer the pan to a wire rack.
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Using a wooden stick, pierce the top in about 20 places and brush the syrup over the cake 4 or 5 times, allowing it to seep in each time.
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Sprinkle with the remaining 1/4 cup walnuts and allow to cool completely, loosen the edges with a knife.
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Cut into 16 squares.