Preheat oven to 325°F. Lightly butter a glass or enamel-coated 11x7-inch baking dish. Place doughnut pieces in baking dish.

Whisk together half-and-half, eggs, sugar, salt, and vanilla in a medium bowl until well combined. Pour mixture over doughnuts in baking dish; let sit 1 hour.

Place baking dish inside of a metal 9x13 baking dish. Fill 9x13 dish with water, allowing water to come halfway up the sides of the 11x7 dish, creating a water bath. Place water bath in preheated oven and bake 40-43 minutes, or until custard is just set. Remove from oven and let cool in water bath until just cool enough to handle, about 30 minutes. Remove 11x7 dish from water bath to scoop and serve. 

Butter Bourbon Sauce: Stir together sugar, water, and corn syrup in a medium heavy-bottomed saucepan. Cook, stirring constantly, over medium until sugar dissolves, about 2 minutes. Increase heat to medium high and cook, occasionally brushing sides of saucepan with a wet pastry brush, until mixture darkens to a deep amber color, about 6-7 minutes. Remove from heat and pour in cream (mixture will bubble vigorously); stir in butter, bourbon, and salt until well combined. Let cool 5 minutes, then spoon sauce over Krispy Kreme Bread Pudding and vanilla ice cream, if desired.