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1
Bring chestnuts, milk, sugar, vanilla, and salt to a boil over high heat in a small saucepan. Reduce heat, bring to a simmer, and cook, stirring occasionally, until milk is reduced by one-quarter and chestnuts smash fairly easily against the side of the pot with the back of a spoon, 15-17 minutes.
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2
Meanwhile, finely chop 4 oz. chocolate. Using a peeler, shave remaining 1 oz. chocolate; set aside until ready to serve.
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3
Puree chestnut mixture in a blender on high speed until very smooth, about 2 minutes. Add chopped chocolate and 1 tsp. brandy and blend on high speed until chocolate is melted, about 30 seconds. Scrape into a large bowl and let cool to room temperature, stirring occasionally (if lumps form, stir vigorously until smooth), 15-20 minutes.
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4
Whip 2 cups cream with a whisk in another large bowl until it just barely holds soft peaks. Stir one-third of cream into chocolate mixture to lighten it, then fold in remaining cream just until incorporated. Transfer mousse to a large serving bowl, cover, and chill at least 2 hours or up to 1 day.
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5
Whip remaining 1 cup cream and 1/2 tsp. brandy with whisk in a large bowl to medium peaks. Garnish mousse with reserved chocolate shavings and serve with whipped cream alongside.
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6
Mousse can be made 1 day ahead; cover and chill.
