Preheat the oven to 350°F. Position a rack in the center.
For the bread pudding: Spread the bread cubes on a large baking sheet in a single layer. Bake for 10–15 minutes, stirring once, until the cubes feel dry and firm with no browning. Cool completely.
Lightly grease a 9×9-inch baking pan with cooking spray.
In a large bowl, whisk together the eggs and brown sugar until smooth. Whisk in the heavy cream, half-and-half, vanilla extract, almond extract, cinnamon, nutmeg and salt.
Add the dried, cooled bread to the custard. Press down so every cube is submerged. Soak for 30 minutes, stirring occasionally.
Transfer the mixture to the prepared pan and smooth the top. Cover tightly with foil and bake on the middle rack for 45 minutes. Remove the foil and bake for 15–20 minutes more, until a knife inserted in the center comes out clean. If the top browns too quickly after uncovering, tent loosely with foil. Let rest for 10–15 minutes.
For the vanilla sauce: In a medium heavy-bottomed saucepan, whisk together the milk, heavy cream, granulated sugar and cornstarch until smooth. Cook over medium-high heat, whisking, until the mixture comes to a gentle simmer. Continue whisking for 1–2 minutes, until lightly thickened and glossy.
Remove from the heat and whisk in the butter and vanilla bean paste. Serve the vanilla bean sauce over the warm bread pudding.