In a large mixing bowl, whisk together, milk, sugar, vanilla, cinnamon and nutmeg with eggs until fully combined. Add cubed doughnuts. Let sit for 15-20 minutes or until most of the custard has been absorbed. (Cut each doughnut into 8 pieces)
To Bake: Preheat oven to 350°F. Spray a 3 quart round or rectangular 13 x 9 inch baking dish with cooking spray.
Pour doughnut mixture evenly into baking dish. Bake for 35-45 minutes until puffed and lightly golden. Let stand for 5 minutes.
To Make Buttered Rum Sauce: Meanwhile, Over medium-high heat, melt butter in a small saucepan. Add sugar, stirring until dissolved. Whisk in whipping cream, corn syrup, rum, and cinnamon. Heat while stirring until there's a gentle bubbling, then lower heat and simmer for 3-5 minutes. Keep warm until serving.
Serve topped with buttered rum sauce and a scoop of vanilla ice cream.