Tomato sauce is the grand sauce built from a stock plus tomatoes, with roux optional. The stock provides the body and depth, while tomatoes give the characteristic color and flavor. Some versions thicken with a little roux, but that step isn’t required, which matches the “roux optional” idea. The other grand sauces rely on milk or white/brown stock thickened with roux, or in the case of Hollandaise, an egg-yolk and butter emulsion, not tomatoes. So the sauce made from a stock and tomatoes is the Tomato Sauce.

The idea is using a flavored fat to finish dishes, letting the fat melt over hot meat or into a finished sauce to deposit aroma and richness. This is done by blending softened butter with flavorings such as herbs, garlic, citrus zest, or spices, then shaping and chilling it. When you plate or finish a dish, a slice or dollop of this flavored butter melts on contact, distributing flavor evenly and giving a glossy finish. This specific term is compound butter. Roux, by contrast, is a thickening mixture of fat and flour used to thicken sauces and soups, not to finish with added aromatics. Beurre manié is butter kneaded with flour used to thicken sauces quickly, rather than to finish with pure flavor components. Liaison is a mixture of egg yolk and cream used to enrich and lightly thicken sauces.

Simmering with the lid just cracked keeps the liquid at a gentle, controlled heat without a full boil. This balance lets flavor particles and gelatin be extracted without whipping the surface into a froth that would cloud the stock. The slight opening lets steam escape, so you don’t lose too much water and you avoid boil-overs, while condensation that might drip back into the pot is minimized. At the same time, impurities and scum can rise to the surface and be skimmed off, helping maintain a clear stock. So this approach preserves volume and clarity while still extracting flavor.

Paysanne is a knife-cut that yields thin, flat pieces of vegetables. The defining trait is the flat shape and small size, which lets the pieces cook quickly and evenly. A description of about 1/8 inch thick with the longest dimension around 1–2 inches matches how paysanne pieces are typically cut, giving a rustic, slicey shard that integrates well into soups, stews, or garnishes. This is different from rounds (coins) or long sticks or rigid cubes, which represent other cuts and cooking needs.