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Preheat oven to 350.
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Grease the bottom and 1/2 inch up the side of an 8x8x2 inch baking pan.
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In a small bowl, stir together the 1/3 cup granulated sugar, the 2 tablespoons coffee crystals, and cinnamon.
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For batter: In a large bowl, stir together flour, 1/2 cup granulated sugar, baking powder, baking soda, and salt.
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In another bowl, combine egg, yogurt, melted butter, and vanilla. Add yogurt mixture to flour mixture; stir until moistened.
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Spread half of the batter in prepared pan. Sprinkle with half of the coffee mixture.
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Repeat layers. Using a knife, swirl batter slightly. Sprinkle with walnuts.
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Bake about 30 minutes or until a toothpick inserted in the center comes out clean.
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For icing: In a small bowl, stir the 2 tsp coffee crystals into the milk until dissolved. Stir in powdered sugar until smooth.
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Drizzle over coffee cake. Serve warm.