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1
Heat the oven to 350 degrees F and arrange a rack in the middle.
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2
Melt 1 stick of the butter in a cast-iron skillet over medium heat.
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3
Add the brown sugar and chopped ginger and cook, whisking occasionally, until the mixture is bubbling, about 4 minutes.
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4
Place the pears cut-side down in the skillet, arranging them in a circle with the stem ends pointing toward the center.
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5
Remove from the heat and set aside.
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6
Whisk the flour, baking powder, baking soda, salt, cinnamon, and ground ginger in a large bowl to aerate and break up any lumps; set aside.
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7
Place the remaining stick of butter and granulated sugar in the bowl of a stand mixer fitted with a paddle attachment.
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8
Beat on medium speed until light and fluffy, about 3 minutes.
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9
Stop the mixer and scrape down the paddle and the sides of the bowl with a rubber spatula.
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10
Add the eggs and vanilla and beat on medium speed until incorporated, about 1 minute.
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11
Stop the mixer and scrape down the paddle and the sides of the bowl with the rubber spatula.
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12
Add a third of the reserved flour mixture, turn the mixer to low, and mix until the flour is just incorporated.
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13
Add half of the milk and mix until just incorporated.
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14
Continue with the remaining flour mixture and milk, alternating between each and ending with the flour, until all of the ingredients are incorporated and smooth.
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15
Stop the mixer, remove the bowl, and stir in any remaining flour streaks by hand, making sure to scrape the bottom of the bowl.
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16
Dollop the batter over the pears and smooth it into an even layer, leaving about a 1/2-inch border from the edges of the pan (the batter may not totally cover the pears).
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17
Bake until the top and edges of the cake are golden brown and a cake tester or toothpick inserted into the center comes out clean (test several spots because you may hit a pear), about 30 to 35 minutes.
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18
Transfer the skillet to a wire rack and let it cool for 20 minutes.
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19
Run a thin knife around the perimeter of the skillet and invert the cake onto a large serving platter (if any pears fall out, just arrange them back into the cake).
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20
Cut into wedges and serve.