In a saucepan combine butter, milk, corn syrup, and vanilla over medium heat stirring until butter is melted. Reduce heat to low, add the chocolate, and stir until smooth.
Remove from heat and quickly whisk in icing sugar until smooth. If too thick, add a bit more milk until mixture is thin enough to dip (rewarm mixture over stove if necessary).
Invert a doughnut into the dip, pressing until it's halfway covered, lift up taking care not to drip down the side. Place on a wire rack for 1-2 hours to allow the glaze to set and harden. While dipping, if glaze is setting up too fast in saucepan, return to heat for a few minutes to soften, or place saucepan over a bowl of hot water.