Preheat the oven to 350 degrees F. Thoroughly grease a 12-cup bundt pan with the butter, making sure to cover the entire inner surface. Sprinkle the flour over the butter and shake the pan to create an even layer of flour. Shake out any excess.
Whisk together the cake mix, ice cream and eggs in a large bowl until well combined; pour into the prepared bundt pan. Bake until a cake tester comes out clean and the sides of the cake are beginning to pull away from the edge of the pan, 35 to 40 minutes. Cool the cake in the pan for 20 minutes, then place a cooling rack over the pan and invert the cake onto the rack. Cool completely.
Microwave the cream cheese in a microwave-safe container in 10 second increments, stirring between each, until pourable. Make the glaze by whisking together the cream cheese, confectioners' sugar and milk in a small bowl until smooth. Drizzle over the top of the cake so the glaze covers the top and runs down the sides.
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