For successful frying in general - with any ingredient - high heat is key. The milk solids in butter will burn at higher temperatures, so a high-heat safe oil like avocado or peanut is the best choice. To fry potatoes in butter successfully, first, you need to clarify the butter. This process removes the milk solids and raises the smoke (burning) point of the butter. First, melt several sticks of butter in a small pot over the lowest heat possible. Once the butter is melted, you will see the white milk solids sitting at the bottom of the pot. Carefully pour or ladle away the translucent milk fat into a heat-proof container. Be careful to leave the solids in the pot and discard them. Store clarified butter, or ghee, in the refrigerator.