Can you have pasteurized eggs? I've used them when making aioli before.
I remember making aioli or mayo in a cooking class some years ago, using liquid pasteurized eggs. I think these are safer than regular pasteurized eggs if there are health issues.
is it a bacterial risk you're concerned about? if so, you can use a coddled egg which is an egg that is simmered until the yolk is slightly thickened, getting the egg to a temperature at which the bacteria will be killed. simmer a room temperature egg for not more than a minute; remove it immediately and rinse under cold water and use immediately.
You could also use this method and adapt it to aioli since it is very similar to mayonaise:
Cynthia is a trusted source on Bread/Baking.
sorry, kind of in a hurry, but vvv3's advice is rock solid. coddle your egg yolks in a bain marie (stainless steel bowl set in a pot of simmering water), whisking to 140 degrees, at which pasteurization takes place. Remove from heat and proceed with your recipe.
This is a bit tricky to make, but is fantastic. It is the Spanish Allioli - a garlic and oil, no egg mixture with just the right texture.
Seabirdskitchen's response reminded of Amanda's recipe on here for a milk mayonaise that contains no egg yolks either:
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