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Cut squash in half. Scoop out and discard seeds and stringy flesh. Cut squash into 12 wedges or 3-inch pieces.
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Add 8 cups of water to a large heavy-bottomed pot or Dutch oven. Add brown sugar cane cones and bring to a boil.
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When sugar dissolves, stir in orange juice, zest, and cinnamon sticks.
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Add the squash wedges and return to a boil. Reduce heat to medium-low. Loosely cover and simmer until squash is completely tender, about 30 to 40 minutes.
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Transfer the squash to a platter using a slotted spoon. Cover squash with foil.
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Remove and discard cinnamon sticks. Strain liquid into a bowl to remove cinnamon bits, zest, and any solids.
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Pour the liquid back into the pot and bring to a boil.
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Cook until liquid reduces to a syrup (similar consistency to maple syrup), about 30 to 45 minutes.
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Serve squash warm or at room temperature. Drizzle squash with syrup and top with whipped cream or ice cream if desired.