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Lightly grease a 9" x 13" pan; line the bottom of the pan with parchment if desired.
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To make the cake bars: In a small bowl, combine the yeast and milk. Set aside.
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In a large mixing bowl or the bowl of a stand mixer fitted with the flat beater attachment, beat together the butter, sugar, and salt until combined and smooth. Scrape the bowl, then beat in the eggs and vanilla bean paste.
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Add the flour to the mixing bowl in three additions, alternating with the yeast mixture in two additions. Stir briefly between each addition to combine, then mix the batter well, stopping to scrape the bowl once more. Once all the ingredients are combined, mix at medium speed to achieve a thick, smooth batter, 1 to 2 minutes.
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Dollop the batter into the prepared pan in 4 or 5 mounds, then spread it evenly across the bottom in a thin layer. (A small offset spatula is a helpful tool here.) Cover and set aside to rise for 1 hour. The dough won’t rise much but should look puffy.
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Towards the end of the rising time, preheat the oven to 350°F.
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To make the topping: In a small bowl, whisk together the corn syrup, water, and vanilla bean paste.
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In a large mixing bowl or the bowl of a stand mixer fitted with the flat beater attachment, beat (“cream”) together the butter, sugar, and salt until light and fluffy, about 4 to 6 minutes on medium speed. Scrape the bowl, then mix in the egg until combined.
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Add the flour in three additions, alternating with the corn syrup mixture in two additions. Scrape the sides of the bowl and mix for 30 seconds more, until well combined.
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Spoon the topping over the risen cake layer in 4 or 5 mounds, then use an offset spatula to gently spread it evenly across the cake layer.
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Bake the butter cake bars for 40 to 45 minutes, until lightly browned; the edges will become darker than the center. The cake will rise and fall in waves, become wrinkled at the edges, and will still be liquid in the center when done. (The length of the baking time is a key part of this cake bars recipe; the more liquid the center remains, the gooier your cake bars will be!)
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Remove the butter cake bars from the oven and cool completely in the pan. Dust with confectioners' sugar or snow white non-melting sugar before serving at room temperature.
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Storage information: Store any leftover butter cake bars, well-wrapped, in the refrigerator for several days. Freeze for longer storage.