Why it happens
Mayonnaise is an emulsion, tiny droplets of oil held apart in egg yolk. If the oil goes in too fast, or the ingredients are very cold, the yolk can't coat the droplets quickly enough and they join back together, so the sauce splits.
Whisk a fresh egg yolk in a clean bowl, then slowly whisk the split mayonnaise into it a little at a time until it thickens again.
Mayonnaise is an emulsion, tiny droplets of oil held apart in egg yolk. If the oil goes in too fast, or the ingredients are very cold, the yolk can't coat the droplets quickly enough and they join back together, so the sauce splits.