Corina's recipe makes the aioli in a mortar and pestle. I make it in a bowl. I also start with 1/4 cup of a neutral oil (like safflower or canola) before adding up to 3/4 cups of olive oil. I think it's easier to emulsify the oil and egg with the neutral oil. The first drops of oil are key. Literally add a drop at a time, while whisking to get the oil and egg to marry. A little dijon added to the egg yolk at the beginning will also aid the emulsification.

makes 1 cup

1 cloves garlic

1/2 t. salt

1 egg yolk

few drops water

3/4-1 C. extra virgin olive oil (or 1/4 c neutral oil and 3/4 c olive oil)

pinch of cayenne

lemon

with a mortar and pestle pound the garlic with salt to a smooth paste. 

add the egg yolk

add a few drops of water

whisk to an even smooth texture.

slowly, very slowly, drip by drop whisk in the olive oil. it should get thick. like mayonnaise.

add a pinch of cayenne

add a few drops of fresh lemon

taste for salt and balance

bread or boiled potato can be used to help stabilize the sauce

if it breaks… start with a fresh egg yolk and whisk the broken sauce slowly into the egg…