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Bring a pot of salt water to a boil. Add the pumpkin pieces and boil for 5 minutes.
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Drain the pumpkin and set aside.
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Peel and chop the shallots into rings.
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Add the butter and the olive oil to the pan.
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Add the shallot rings and cook, stirring frequently until they turn golden.
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Add the pumpkin pieces and the chestnuts to the pan and stir to combine.
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Add the fresh thyme and the vegetable broth to the pan.
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Fry for 15 minutes, stirring frequently.
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Season with salt and freshly ground pepper to taste.
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Stir in the ground nutmeg.
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Remove from the heat, transfer to a serving dish and serve.