Sauce Architecture: Building a {sauce_type} Sauce from Scratch

You can follow a sauce recipe but you do not understand it. Name the sauce, beurre blanc or mole or whatever, and it teaches the structural logic: fat, liquid, emulsifier, aromatics, acid, seasoning, what each part does and the decisions that make or break it. Aimed at cooks who want to build sauces, not just copy them.

Teach me to make a {sauce_type} sauce (e.g. beurre blanc, velouté, mole, XO) completely from first principles. I am an advanced home cook who understands basic technique but wants to understand the structural logic behind this sauce, not just follow a recipe. Explain the foundational components (fat, liquid, emulsifier, aromatics, acid, seasoning agents), how each contributes to texture and flavour, and the key decision points where the sauce can succeed or fail. Walk me through the classic method, then explain two deliberate variations I can make by changing one variable (e.g. the fat type or the acid source) and what the result will be. Include troubleshooting guidance for the two most common failures specific to this sauce type.