Preheat oven to 350°F (175°C) with a rack in the center. Cut the day-old brioche or challah into 1-inch cubes, leaving the crusts on, and spread in a single layer on a rimmed baking sheet.
Bake for 10 to 12 minutes, tossing once halfway through, until the cubes feel dry and look lightly golden. Cool for 5 minutes and keep the oven on.
Cover the golden raisins with very hot tap water and let sit for 10 minutes, then drain well and pat dry. Grease the bottom and sides of a 9x13 baking dish with the 1 tablespoon softened butter, reaching the corners.
In a large bowl, whisk the 4 eggs until fully blended. Add the 3/4 cup granulated sugar and 1/4 cup brown sugar and whisk for about 1 minute until slightly lighter.
Pour in the 2 cups whole milk, 1 cup heavy cream, and 2 tablespoons melted butter, whisking as you go. Stir in the vanilla, cinnamon, nutmeg, and salt until smooth with no clumps.
Add the toasted bread cubes and drained raisins to the custard and fold gently with a rubber spatula so you do not crush the bread. Press the cubes down lightly so every piece touches the liquid.
Let the bowl rest for 15 minutes, folding once or twice, until most of the custard has been absorbed. The bread should look saturated but not completely mushy.
Spoon the soaked bread mixture into the buttered 9x13 dish and spread into an even layer, pouring any leftover custard over the top. Sprinkle the 2 tablespoons turbinado sugar evenly over the surface for a crisp, sparkly crust.
Bake uncovered for 45 to 55 minutes, until the top is puffed and deep golden and the center barely jiggles. If the top browns too fast, cover loosely with foil after 30 minutes.
Check the center with an instant-read thermometer; it should read about 160°F (71°C), or a knife inserted in the middle should come out clean.
While the pudding bakes, melt the 1/2 cup butter in a medium saucepan over medium-low heat. Stir in the 1/2 cup granulated sugar and 1/2 cup heavy cream and simmer gently for 4 to 5 minutes, stirring often, until the sauce thickens slightly and coats the back of a spoon.
Remove the pan from the heat and stir in the 2 teaspoons vanilla extract and a pinch of salt. Keep the sauce warm; it will thicken a little as it cools and can be rewarmed for 20 seconds in the microwave if needed.
Move the baked bread pudding to a wire rack and let it rest for 10 to 15 minutes so the custard finishes setting and the squares hold their shape. Cut into 12 squares with a sharp knife.
Spoon a generous amount of warm vanilla butter sauce over each portion and serve right away while the edges are crisp and the center is soft.