In a small sauté pan, lightly cook the garlic in 1/8-1/4 c. of the olive oil. Do not let the garlic brown, just soften; about one minute.
On medium-low speed, mix the egg yolk and then slowly add some olive oil, until you have a nice base, a couple of tablespoons worth. DO NOT add the oil too quickly; start with a drop at a time and as the aioli gets lighter in color and thicker, you can increase to a very fine stream.
Then add the lemon juice, vinegar, mustard, and salt until well blended and light yellow, about one minute.
With mixer on medium-low, slowly drizzle in the garlic and its cooking oil into the egg mixture. When the oil is completely mixed in, you can continue to add oil slowly until it is the desired consistency.
If you have an immversion blender with a whisk attachment, it works better than a stand mixer when used in a small bowl... Just don't go quite so slow as described above or your aioli will be too fluffy and not so creamy.
Goes great with our oven fries recipe!