Doughnut Pudding

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Ingredients

butter for the pan
6 cups day-old raised doughnuts cut into 1-inch pieces (from 4 or so glazed doughnuts or apple fritters)
4 large eggs
2 tablespoons dark rum or 1/2 teaspoon rum extract (optional)
1/2 teaspoon ground cinnamon
1/4 cup granulated sugar
1 teaspoon vanilla extract
1 cup whole milk
3/4 cup heavy (whipping) cream
1 cup confectioners’ sugar
1 teaspoon vanilla extract
1 1/2 tablespoons hot water