Many sauce and soup recipes need to be reduced and thickened, which means gently simmering to achieve the desired consistency.
With sauces and soups containing milk, boiling or simmering can cause the milk to curdle, which is not particularly appetizing (although it is safe to eat).
Without getting too complicated, milk is a mixture (called an emulsion) of butterfat, proteins, and water. What happens when you boil it is that the three components of the emulsion break apart: the milk proteins coagulate and separate from the water, producing what is commonly known as curdled milk.
This is how cheese is made: the milk solids are coagulated through cooking and the addition of an enzyme called rennet, and then the excess liquid is drained away. (If you ever see drops of oil coming off your melted cheese, that's also because of the emulsion breaking, and it usually happens because it's a low-moisture variety of cheese.)
But for your sauce or soup, you don't want curdled milk, you want it to be nice and smooth. So here are some tips to help prevent milk from curdling when you heat it:
Don't Let it Boil
Boiling is a sure way to curdle milk. But it's not just boiling. Heating milk too quickly, even if it never comes to a boil, can also curdle it. Instead, heat the milk gently over medium-low heat.
Stabilize with a Starch
Starches like flour or cornstarch help stabilize the milk emulsion and prevent it from separating. A common technique is to thicken your sauce or soup with roux before adding the milk.
Avoid Strong Acids
If your sauce or soup contains an acidic ingredient like wine, tomatoes or lemon juice, the milk is more likely to curdle. To counteract the effect of the acid, you can use a starch as described above.
Season at the End
Salt is another ingredient that can cause milk to curdle. But, obviously, you need to season your sauce. The key is to add the salt at the end, rather than cooking or reducing it with the salt already in it. (Seasoning your sauces and soups at the very end is a good habit to get into anyway.)
Temper the Milk
Don't add cold milk directly into a hot liquid. Instead, whisk small amounts of the hot liquid into the cold milk. When the milk is warm, then add it into the hot liquid. This process is called tempering. Or, simply heat the milk gently in a saucepan before adding it.
Use Cream Instead
Dairy products with higher fat contents, such as whipping cream and heavy cream, are less prone to curdling. Restaurants use heavy cream for making sauces and soups because unlike milk, it can be boiled without curdling. (It also has more flavor and richness than milk.) Conversely, 2% milk is more likely to curdle than whole milk.