Stocks and the five mother sauces: bechamel, veloute, espagnole, hollandaise, tomato. The foundation under almost every classical French preparation and most modern Western cooking.
Bechamel is the simplest of the five mother sauces. The technique is two steps: 1. **Roux.** Equal weights of butter and flour cooked together to form a paste. 2. **Liquefy.** Hot milk whisked in gradually over heat until smooth and thickened.
Espagnole is the most demanding of the five mother sauces. It is built on a brown roux (cooked further than the blond roux of veloute), made with brown stock (roasted bones), thickened with vegetables, and seasoned with tomato puree. The base is then simmered for hours to deepen, strained, and often reduced further into demi-glace, a syrupy concentrated meat reduction.
Hollandaise is the simplest of the mother sauces in ingredients, the trickiest in technique. It is an emulsion: tiny droplets of fat (the butter) suspended in a stable matrix held by an emulsifier (the egg yolks' lecithin). Add the butter too fast or let the eggs overheat, and the emulsion breaks: the sauce separates into a greasy puddle with a curdled scramble at the bottom.
Stock is not soup. Stock is what soup is made from. The difference is seasoning: stock is unseasoned (or barely seasoned), so it can be reduced, salted, and finished into whatever final dish you want. Soup is salted, often thickened, and ready to eat.
Classical French cuisine is built on stocks and the five mother sauces. Auguste Escoffier codified the system at the turn of the 20th century. The system has held up for more than a hundred years because it works: a small number of foundational recipes, each of which is the parent of dozens of derivatives. Once you can make a clean stock and the five mothers, you can make hundreds of finished sauces by changing one or two ingredients.
Tomato sauce occupies a strange place in classical French canon. Escoffier added it to the five mothers in the early 1900s, recognising its ubiquity in Mediterranean and Italian-French cooking. Some French purists still treat it as a daughter sauce; the rest of the world treats it as foundational.
Veloute is bechamel's stock-based twin. The technique is identical: cook a roux, then whisk in warm liquid until thickened. The difference is the liquid: hot stock instead of hot milk. The result is a sauce that tastes of the protein the stock came from, with a smooth velvety mouthfeel.