You were whisking, it looked glossy and pale, and then it turned into a greasy puddle with little flecks floating in clear yellow fat. Annoying, but your sauce is almost always fixable, and the cause is usually one of a handful of simple things.
Beurre blanc is an emulsion: tiny droplets of butterfat suspended in a small amount of acidic liquid (usually a reduction of white wine, vinegar, and shallots). Butter already contains water and milk solids that help hold that suspension together. When the balance between fat, water, and temperature tips too far, the droplets merge and the fat separates.
This is the number one cause. Once beurre blanc goes much above about 140°F (60°C), the butter melts completely instead of softening into the sauce, and the emulsion breaks. A sauce that bubbles even briefly is in danger.
Signs: the sauce suddenly thins, looks oily, and fat pools at the edges.
Dumping in half a stick at once overwhelms the small amount of liquid. The sauce can only absorb fat gradually. Add one or two cubes at a time and wait until each is almost fully incorporated before adding more.
Your reduction should end up as about 2 tablespoons of syrupy liquid, not a dry layer of shallots. Without enough water, there's nothing for the fat to be suspended in.
Cold butter melts slowly, which gives it time to emulsify. Room-temperature or soft butter melts all at once and behaves like pouring in oil. Cut it into small cubes and keep it in the fridge until you need it.
A beurre blanc left on the counter cools, the butter firms up, and when you reheat it, it often breaks. If it gets too cold, it can also turn grainy and stiff.
If it's just starting to look oily: take the pan off the heat immediately. Whisk in an ice cube or a tablespoon of cold water, or a cube of cold butter. The sudden drop in temperature often pulls it back together within seconds.
If it's fully broken: don't keep whisking the broken sauce in the same pan. Instead: 1. In a clean pan or bowl, put about 1 tablespoon of cold water, cream, or a bit of leftover reduction. 2. Over very low heat (or none), whisk the broken sauce into it a teaspoon at a time at first. 3. Once it starts to thicken and look creamy again, you can add the rest a little faster.
This works because you're rebuilding the emulsion from scratch, giving the fat a fresh base to cling to.
If it's too cold and stiff: set the pan over very gentle heat and whisk constantly, just until it loosens. Pull it off the moment it turns smooth.
This article covers the main causes and fixes for beurre blanc. The guide Broken Sauce Rescue: Fix a Split Beurre Blanc or Runny Hollandaise in 2 Minutes (and Stop It From Happening) goes further: it also covers hollandaise, which breaks for somewhat different reasons, and puts the rescue methods into short step-by-step formats you can keep next to the stove. It's a practical reference for people who make these sauces often, but if you only needed to save tonight's dinner, the steps above should be enough to get you there.
Go further : Broken Sauce Rescue: Fix a Split Beurre Blanc or Runny Hollandaise in 2 Minutes (and Stop It From Happening (5.00 $)