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Preheat the oven to 350°F. Grease an 8" square pan, or line it with parchment or a reusable silicone liner; grease the parchment or liner.
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To make the pumpkin spice bars: In a medium saucepan (one with a 2-quart capacity works well), combine the sugar, pumpkin purée, oil, butter, pumpkin pie spice, and salt. Cook over medium heat, stirring constantly with a spatula, until the butter melts, the sugar is dissolved, and the mixture is smooth, 3 to 5 minutes.
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Remove the saucepan from the heat and set it aside to cool for 10 to 15 minutes, stirring occasionally.
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Add the egg and stir to combine. Add the flour and baking powder and continue to stir until the batter is smooth and there are no visible streaks of flour. Transfer the batter to the prepared pan and smooth it into an even layer.
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Bake the pumpkin spice bars for 32 to 36 minutes, until they appear set across the top and a toothpick inserted into the center comes out clean. Remove them from the oven and let cool for 15 minutes before loosening the edges and transferring the bars out of the pan onto a rack to cool completely.
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To make the espresso glaze: In a medium microwave-safe bowl, combine the milk, espresso powder, and salt. Heat in 10-second increments, stirring between each, until the espresso powder is dissolved.
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Add the confectioners’ sugar to the milk mixture and stir until smooth. Pour the glaze over the cooled bars and spread it to the edges. Allow the glaze to set before cutting and serving.
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Storage information: Store any leftover pumpkin spice bars, covered, at room temperature for several days. Freeze the unglazed bars for up to 2 months; freezing the glazed bars is not recommended.
