Add the cubed white bread and raisins to a large mixing bowl. Set aside to let the raisins soften slightly.
In a 1-quart saucepan, heat the milk and 1/4 cup of butter over medium until the butter melts completely—about 4 to 7 minutes. Do not let it boil.
Pour the warm milk mixture over the bread and raisins. Gently stir to coat, then let sit for 10 minutes to absorb the liquid.
Stir in the sugar, slightly beaten eggs, vanilla, and ground nutmeg until just combined. Keep the mixture slightly rustic—don’t overmix.
Preheat oven to 350°F. Grease a 1½-quart casserole dish and pour the mixture in. Bake uncovered for 40 to 50 minutes until the center is set and the top turns golden. A knife inserted should come out clean.
In a 1-quart saucepan, combine butter, white sugar, brown sugar, and heavy cream. Cook over medium heat for 5 to 8 minutes, stirring occasionally, until thickened and boiling. Remove from heat and stir in vanilla.
Spoon the warm bread pudding into bowls and top with the hot vanilla sauce.