Mince and mash garlic to a paste with a pinch of salt using a large heavy knife.
Whisk together yolk, lemon juice, and mustard in a bowl.
Combine oils and add a few drops at a time to yolk mixture, whisking constantly, until all oil is incorporated and mixture is emulsified. (If mixture separates, stop adding oil and continue whisking until mixture comes together, then resume adding oil.)
Whisk in garlic paste and season with salt and pepper.
If aioli is too thick, whisk in 1 or 2 drops of water. Chill, covered, until ready to use.
Serve with carrot and celery sticks.
