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1
Preheat the oven to 350u00b0F, and put the kettle on to boil.
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2
For the base, process the crackers, butter and heaped tablespoon chestnut puree until like fine crumbs. Press the mixture into the bottom of a 9 inch springform pan and place in the fridge while you make the filling.
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3
Beat the cream cheese until smooth and add the sugar. Add the eggs and egg yolks, beating them in one by one until they are incorporated into the cream cheese and sugar. Pour in the sour cream, lime juice, vanilla extract and rum, and beat again until smooth and creamy. Finally fold in the sweetened chestnut puree. Don't worry about making a fully amalgamated mixture: smooth cream cheese with grainy streaks of chestnut is just fine.
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4
Line the outside of the springform pan containing the crumb base with a good wrapping of plastic wrap, so that the whole of the bottom and sides are enveloped in plastic. Do the same with aluminum foil, covering the layer of plastic wrap to make a very watertight casing. Stand the springform, thus covered, in a roasting pan and pour in the chestnut filling. Once that's done, pour water from a recently boiled kettle into the roasting pan to come just over an inch up the side of the pan (the plastic wrap will make it bob up and down a bit) and place in the oven to cook for an hour.
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5
When the cheesecake's ready it should be just set on top with a hint of wobble underneath; it certainly carries on cooking as it cools. Take the cheesecake out of the roasting pan, take off the foil and plastic wrap and let the cheesecake cool on a rack. Refrigerate overnight before unmolding and leaving it to get back to room temperature. If you need to unmold it long before you want to eat it (I often do just because I like to get all bothersome stuff out of the way before people arrive) then just sit it on its serving plate in the fridge until about 20-30 minutes before you want to eat it. I'd take it out as you sit down to dinner or lunch or whatever.
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6
You can make the syrup in advance but do not pour over until the actual point of serving.
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7
You just put all of the syrup ingredients into a saucepan and melt together. Let the syrup boil for 10 minutes, then cool to just warmish (or even room temperature) before criss-crossing the top of the cheesecake with it.
