Melt the butter in a small sauté pan over medium-low heat. Add the garlic, and cook until softened but not browned, 1 to 2 minutes. Add the anchovies and oil, and reduce heat to low. Cook, stirring occasionally, until the mixture is softened and lightly browned, about 20 minutes. Remove from heat; let cool slightly.
Transfer the mixture to the jar of a blender, and add the milk; puree until thick and smooth, about 2 minutes. Transfer to an airtight container, and store in the refrigerator until ready to serve, preferably overnight and up to 4 days.
