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To prepare the pudding, combine the brown sugar, water, and salt in a large heavy-bottomed pot over medium-high heat.
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Cook to a smoking, dark caramel, about 10 to 12 minutes. (The sugar will smell caramelized and nutty and turn a deep brown.)
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Immediately whisk the cream and milk into the caramel to stop the cooking. The mixture will steam and the caramel will seize, but will become smooth again as you whisk.
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Bring the mixture to a boil and reduce the heat to medium.
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In a medium bowl, whisk together the egg, egg yolks, and cornstarch.
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Temper the hot caramel cream into the egg mixture by adding a cup of caramel at a time, whisking constantly. When half the caramel is incorporated, pour the egg mixture back into the remaining caramel, and boil while whisking constantly until the custard is very thick, about 2 minutes.
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Remove the custard from the heat and whisk in the butter and rum.
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Strain the custard through a fine mesh strainer to remove any lumps and divide among ten 6-ounce ramekins.
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Cover with plastic wrap and chill for several hours, or up to 3 days.