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1
Preheat oven to 350 degrees F (175 degrees C). Place squash, pointed-side up, on a baking sheet.
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2
Bake in the preheated oven until squash is tender, about 1 hour. Cool squash until easily handled. Cut squash in half, remove seeds and stringy parts of meat, and peel skin from the squash and discard.
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3
Mash the squash meat in a bowl and measure 1 cup meat.
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4
Combine 1 cup squash meat and butter in a saucepan; add enough water to make squash stir-able. Cook squash mixture over low heat until squash softens, adding more water if needed to thin the squash, about 15 minutes. Remove saucepan from heat and cool squash mixture.
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5
Beat egg yolks and brown sugar together in a bowl using a whisk until thickened.
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6
Heat cream and milk together in a saucepan over low heat until mixture begins to simmer. Gradually pour cream mixture into the egg yolk mixture and beat until smooth to temper the eggs. Pour the cream-egg mixture into the saucepan; simmer over low heat until mixture is thickened and easily coats the back of a spoon, about 20 minutes. Strain cream mixture through a fine-mesh strainer into a large bowl and cool to room temperature.
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7
Beat squash mixture into cream mixture until smooth; stir in vanilla extract, ginger, salt, and allspice. Cover bowl with plastic wrap and refrigerate until completely chilled, at least 4 hours.
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8
Stir cream mixture and transfer to an ice cream maker and follow manufacturer's instructions for making ice cream.
