1. Put the anchovies in a bowl, cover with cold water, and let soak for 30 minutes. Drain and pull the anchovies into halves, discarding the bones and setting the fillets aside.

2. Separate the heads of garlic into cloves, but do not peel them. Place the whole, unpeeled garlic cloves into a small saucepan and add the milk. Bring to a boil over medium-high heat, then lower to a simmer and simmer until the cloves are tender, about 15 minutes. Drain, discarding the milk.

3. Pass the garlic and anchovy fillets through the widest setting of a food mill. If you do not have a food mill, peel the garlic before boiling in milk. Once the garlic is cooked, mash the garlic and anchovies together in a mortar and pestle into a chunky paste.

4. Transfer the garlic-anchovy paste to a medium pot set on a diffuser over very low heat, and add the olive oil and butter. If you do not have a diffuser but have an electric stove, just use the lowest setting on your electric stove. If you have a gas range but no diffuser, set a cast-iron frying pan directly on the burner then put the saucepan on top. Cook for at least 30 minutes or up to 2 hours. The mixture should never bubble or brown; if it does, lower the heat. Season with salt. 

5. The bagna cauda is now ready to use. If serving as a dip for vegetables, pour the sauce into a fondue or bagna cauda pot and set over a candle to keep warm. Can be made a day ahead and gently reheated before using. Will keep, refrigerated, for a week.