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Heat the milk until steaming, either over the stove or in the microwave. Add the jasmine tea leaves, then let steep for 30 minutes. Remove the tea leaves, and stir in half of the granulated sugar and 1 pinch of the salt. Set the jasmine milk tea aside.
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Fill a medium pot half full with water, and bring to a boil. Add the wintermelon and cook for 5-10 minutes, or until the wintermelon is very tender when pierced with a fork. It should also look almost translucent in the center. Save ⅓ cup of the wintermelon cooking liquid, then drain the wintermelon. Stir the rest of the granulated sugar, as well as a pinch of salt, into the wintermelon.
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Blend the wintermelon until pureed (it should look like pale applesauce). If your blender needs some help, add a little bit of the cooking liquid. Don’t worry about how much you add, since you’ll be cooking off the water later on anyways.
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Transfer the wintermelon puree back to the pot you boiled it in, then place over low heat for 15-20 minutes. The goal of this step is to get rid of the extra moisture and concentrate the puree, not get it caramelized or brown, so make sure you scrape the pot frequently.
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You’ll know it’s done when it loses its watery layer on top. If you’re not sure, you can pour it into a measuring cup, and it should be about 1 ½ cups of puree when it’s done (more if you had to add water to blend the wintermelon). Doesn’t have to be exactly this amount of puree, but around that ballpark. Let the puree cool slightly.
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Cook the boba according to the package instructions (depending on how long it takes to cook, it might make sense to cook the boba first). Once the boba is chewy and no longer has a hard, powdery inside, drain it. Mix the boba with the last pinch of salt, as well as the brown sugar.
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Mix together the wintermelon puree and jasmine milk tea.
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Divide the boba and any leftover brown sugar between two glasses, then add a few handfuls of ice to each glass. Pour in the wintermelon tea, and give them a stir before enjoying.