Get ready: Wet a kitchen towel and wring out until just damp. Shape the towel into a ring on the counter and nestle a medium bowl in the center. Place the oil in a measuring cup with a spout and set aside.
Season the egg yolk: Using a microplane or the smallest side of a box grater or microplane, finely grate the garlic into the bowl. Add the kosher salt and egg yolk. Whisk briefly combine.
Add the oil: Whisking constantly, add the oil a few drops at a time. As the aioli starts to emulsify, increase the oil addition to a thin, steady stream, continuing to whisk at a brisk pace.
Finish the aioli: Once all oil has been added, whisk a few more times to ensure that the mixture is thoroughly emulsified. Whisk in lemon juice. Taste for salt and adjust accordingly.
Aioli, kept in a tightly sealed container, will keep for up to one week in the fridge.
If your sauce breaks, try vigorously whisking in 1 more egg yolk. If that doesn’t work, try adding a splash or two of ice water and whisk with all your might!