Fill a stockpot or other food-safe container with water and insert an immersion circulator into the water bath. Set the temperature to 190ºF.
Combine the butter, anchovy fillets and garlic in a quart-sized zipper-lock or vacuum-seal bag. Seal the bag, removing any excess air using the water displacement method or a vacuum sealer.
Place the sealed bag in the preheated water bath and let cook for 3 hours, or until you can mash 1 of the garlic cloves by gently squeezing the bag.
Remove the bag from the hot water bath and empty the contents into a wide-mouth Mason jar. Pour the olive oil into the jar. Use an immersion blender to process the mixture into a thick sauce. Serve warm or at room temperature, with radishes for dipping.