Bring the half-and-half and cinnamon stick to a simmer in a medium saucepan over low heat. Whisk together the egg and egg yolks, sugar, cornstarch and salt in a medium bowl. While whisking constantly, slowly pour the hot half-and-half into the egg mixture. Return the mixture to the saucepan and cook over medium heat, whisking, until the mixture just reaches a boil, about 4 minutes. Continue cooking, whisking constantly, until the mixture thickens, about 1 minute more. Remove from the heat and strain through a fine-mesh sieve into a clean bowl for a super silky texture. Stir in the butter and rum.
Pour the warm pudding into 4 ramekins or glasses. Refrigerate until set, about 2 hours.
Bring the sugar and 3 tablespoons water to a boil in a small saucepan over medium-high heat, stirring until the sugar dissolves. Boil to thicken slightly, about 2 minutes. Add the rum and raisins and simmer to mellow the alcohol taste slightly, 2 minutes more. Remove from the heat and swirl in the butter to bring the sauce together. Keep warm.
When ready to serve, place a dollop of sour cream on top of each pudding and spoon hot buttered-rum raisins over each one. Serve.
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