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1
To roast and prepare the peaches: Heat oven to 425 degrees F. Liberally sprinkle cut sides of halved peaches with sugar. Follow with a pinch of salt. Meanwhile, melt butter in cast iron skillet over medium-high heat. Arrange peaches in the skillet cut side down. Without disturbing them, cook until the cut sides begin to brown, about 3 to 5 minutes. (Note - the peaches will throw off juice while they're browning, which will eventually thicken in the oven.) Transfer to the oven and roast for about 10 minutes. Flip, drizzle with butter, and continue roasting another 5 to 10 minutes, or until tender. The total cooking time will depend on the size and ripeness of your peaches. Let cool -- then transfer the contents of your pan into the bowl of a food processor, making sure to scrape up and include any melted butter left behind. Process for a minute or two, or until you have a smooth puree. Set aside and let cool to room temperature. (The peach puree can be prepared up to 2 days in advance, covered, and refrigerated.)
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2
To make the ice cream base: Whisk together the egg yolks and 1/3 cup sugar until the mixture is pale yellow.
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3
Meanwhile, in a large saucepan, whisk together the heavy cream, whole milk, and remaining 1/3 cup sugar. Bring the mixture to a simmer. As soon as it starts to come up the sides of the pan, remove the pan from the heat. Add a little to the egg yolk mixture to warm it, starting with a few drops, then a slow and steady stream, about 1 cup total. Whisk constantly to prevent the egg yolks from curdling. Pour the egg yolk mixture back into the remaining hot milk mixture; whisk together to combine.
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4
Return the custard to the stove and cook it over low heat, stirring constantly with a heatproof spatula, until it thickens enough to coat the spatula, about 7 to 10 minutes. If using an instant read thermometer, it should read around 170 degrees F (77 degrees C). Immediately take the saucepan off the heat, and pour the custard mixture into a large bowl. Stir in the vanilla extract, salt, and mint.
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5
Prepare an ice bath in a large bowl, and in it, place the bowl of custard. Let stand at room temperature for about 1 hour to cool and to let the mint steep. (Taste the custard after about 1/2 hour to see how minty it is -- if it's to your liking, go ahead and discard mint and continue to cool the custard.) Stir the cooled peach puree into the custard, then strain through a fine-mesh sieve into a bowl, pushing down and scraping back and forth with your spatula; you will have just over 4 cups of custard. Discard the mint. Cover the bowl and refrigerate the custard until it's thoroughly chilled, about 3 to 4 hours or overnight. (I sometimes accelerate this chilling process by putting the bowl in the freezer for 45 minutes or so - but set a timer so you don't forget about it. You want it to chill, not freeze!)
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6
To freeze the ice cream: Transfer chilled custard mixture to the bowl of your ice cream mixture, and freeze according to the manufacturer's instructions. Once done, transfer the ice cream into an airtight container, and place a piece of plastic wrap directly over the surface to prevent crystals from forming. Cover and freeze until firm, about 2 hours. Ice cream should keep well for up to 2 weeks.
