Preheat oven to 350ºF. Whisk together half-and-half, milk, granulated sugar, eggs, melted butter, cinnamon, pepper, 2 teaspoons of the vanilla, and 1 teaspoon of the salt in a large bowl. Add bread and raisins; stir to combine. Let stand 15 minutes, stirring occasionally. Transfer to a lightly greased (with cooking spray) 13- x 9-inch baking dish. Bake until golden and set, 45 to 50 minutes. Cool on a wire rack.

Meanwhile, combine rum, brown sugar, 1/2 cup butter, and remaining 1/2 teaspoon salt in a saucepan over medium. Bring to a simmer; cook, stirring occasionally, until slightly thickened, 15 minutes. Remove from heat; stir in cream and remaining 2 teaspoons vanilla. Drizzle 1/2 cup sauce over bread pudding. Serve with remaining sauce.