How to make mayonnaise with a cooked yolk?
Regarding Romanian recipe...

Actually in Romania people tend not to use raw yolks so much. Most often we eat relatively raw yolks just in fried eggs or soft boiled eggs.
In most of recipes the yolks are cooked.

Regarding mayonnaise there are three ways of preparing it: using just raw yolks (most simple), using raw and cooked yolks 50-50, and using just cooked yolks (harder).

For the last one, which helps storing mayonnaise safer and longer,  the eggs are well boiled and then the yolk is removed. After cooling done until lukewarm it is finely crushed (often by passing it through a sieve several times) and then oil is added little by little. Sometime a small quantity of soft mustard is added for creating more chance of success. For one yolk about 100 ml oil can be incorporated (depending yolk size, as well). The mayo resulting it is slight harder then regular one. For making it softer, if you would like, you can add at last bubbled water (like a teaspoon), some lemon juice and even a tablespoon of cream or yogurt. Enjoy! Pofta buna!!

PS: we have some recipes of mayo without yolks, at all. These are for the period when we traditionally fast and no animal product is allowed.