In a medium bowl, whisk the egg yolks and the remaining 1/2 cup of sugar. When the half and half reaches a simmer, slowly pour it into the bowl with the yolks, whisking constantly, until completely incorporated. Transfer the mixture back to the saucepan and cook over medium heat, stirring constantly, until the mixture thickens and coats the back of a wooden spoon (it will register between 170 and 175 F on an instant read thermometer). Pour through the strainer into the bowl with the cream; stir to combine. Stir in the strawberry puree, lime juice, and rum. Chill the custard until very cold, at least 4 hours but preferably overnight.
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