This sauce is very similar to an Aoili. I have a stick blender at home so I make Aoili all the time. It is similar to Mayo but is flavored. There are many variants but a core Aoili has oil, egg and lemon juice usually with garlic as a key flavor.
2 cloves garlic roughly chopped
2 tsp lemon juice
1 c good olive oil
pinch of salt and pepper
Put all the ingredients (except olive oil) into the stick blender cup and use your stick blender to quickly blend the egg and other ingredients (just a pulse or two). Then pour half the olive oil in the cup and emulsify the mixture. Add more olive oil until you get a mixture with the desired consistency. I usually dump all the olive oil in and just blend it. Don't overblend it.
You can add/replace the egg with mustard which will also act as an emulsifier. You can also replace the lemon juice with any kind of acid like vinegar. And you can add an infinite variety of herbs, spices and flavors.
ps. note this Aoili will last a few days in the fridge but is best used immediately.