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First, decide which sorbet you'd like to make. The Simple Freezer Sorbet requires only a food processor and freezer; the Churned Sorbet requires an ice cream maker.
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To make Simple Freezer Sorbet
Combine the water and sugar in a medium-sized saucepan. Bring the mixture to a boil over medium heat, stirring to dissolve the sugar. Simmer for 5 minutes, without stirring. Remove the syrup from the heat, pour it into a bowl, and set in the refrigerator to cool.
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Wash and trim the strawberries.
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Place the strawberries in the work bowl of a food processor. Process until fairly smooth, but not puréed.
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Add the syrup and citrus juice, and pulse briefly, just to combine. The syrup doesn't have to be cold, but it shouldn't be boiling hot, either. Lukewarm or cooler is fine.
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Place the mixture in a shallow pan; an 8" x 8" or 9" round cake pan are both good choices. Place the pan in the freezer. There's no need to cover it.
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After 2 hours, use a fork or spoon to stir it around, bringing the frozen edges into the center. Return to the freezer.
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Continue to stir every hour or so, until the sorbet is nearly as firm as you like. This may be as little as 4 hours total, start to finish; or it may take longer, depending on the temperature of your freezer. Once the sorbet is entirely icy (like a slush drink), you can purée it in a food processor or using a hand blender, if you like. Place in a bowl, cover, and return to the freezer.
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Sorbet should be ready to serve about 4 to 6 hours after you first put it into the freezer. Waiting a couple of hours beyond that will solidify it beyond scoopable. To serve beyond that window, allow sorbet to soften slightly at room temperature; this will only take about 10 minutes. Scoop into dishes and serve.
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To make Churned Sorbet
Combine the water and sugar in a medium-sized saucepan. Bring the mixture to a boil over medium heat, stirring to dissolve the sugar. Simmer for 5 minutes, without stirring. Remove the syrup from the heat, pour it into a bowl, and set in the refrigerator to cool.
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Place the strawberries and citrus juice in a food processor, and purée till completely smooth.
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Press the purée through a fine-mesh strainer, to remove the seeds.
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Combine the seedless purée with the corn syrup and sugar syrup. Chill for 1 hour, or until refrigerator-cold.
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Make the sorbet according to the manufacturer's directions for your ice cream maker. The finished sorbet will be fairly soft; "ripen" it for a couple of hours in the freezer to firm it up, if desired.