Make sauce: In a large pan, bring milk to a simmer over medium-high heat. Set a fine-mesh strainer over a bowl; put bowl in a container of ice water.
In a large bowl, whisk yolks with sugar until thick, about 3 minutes. Gradually whisk in hot milk. Return to pan; cook over low heat, stirring, until slightly thickened and mixture coats the back of a spoon, 3 to 10 minutes. Strain into bowl; stir in rum and vanilla. Let cool, stirring occasionally. Press plastic wrap directly onto the surface; cover and chill for at least 2 hours and up to 2 days.
Make pudding: Preheat oven to 350°F; butter a 1 1/2-quart shallow baking dish. Place dish on a foil-lined baking sheet.
Spread 1 side of each bread slice with butter. Cut bread into 1-inch pieces, transfer to baking dish and sprinkle with raisins.
In a bowl, whisk eggs, milk, cream, sugar and vanilla. Pour over bread. Bake until tip of a knife inserted into center comes out clean, 45 to 55 minutes. Cool for at least 20 minutes. Serve with sauce.
