Preheat oven to 350 degrees F. Place doughnut pieces in a 3-quart baking dish; set aside.
In a large bowl whisk together milk, sweetened condensed milk, coffee, egg yolks, and 1 tsp. of the cinnamon. Pour evenly over doughnuts. Combine granulated sugar and remaining cinnamon. Sprinkle evenly over doughnut mixture.
Bake, uncovered, for 40 to 45 minutes or until a knife inserted near center comes out clean. Cool slightly.
Meanwhile, in a small saucepan melt butter over low heat. Stir in brown sugar and whipping cream. Cook and stir until sugar is dissolved. Remove from heat. Stir in vanilla. Spoon over pudding. If desired, top with sprinkles.