Place the eggs in a bowl and cover with warm water to bring to room temperature.
Paste the garlic using a microplane or very fine grater. You can also use a garlic press, or finely mince the garlic then smash it with the side of your knife until it turns to a paste.
Juice ½ of the lemon. Measure 1 teaspoon of fresh juice and set aside.
Separate the yolk from one egg and discard the white.
Add the egg yolk, the white and yolk of another egg, 1 teaspoon lemon juice, the garlic paste, and salt to a food processor or blender.
Pulse the food processor until the ingredients are evenly blended and frothy.
Add the grapeseed oil and olive oil to a measuring cup. Turn the food processor on and add the oil very slowly (drop by drop, not a stream).
Once the mixture starts to thicken and you can see it emulsifying (it's a consistent mixture and you can't see the oil), then start pouring faster, in a thin stream.
Once all the oil is added, stop the food processor. Check the mixture — the consistency should be similar to mayonnaise. Taste and add more salt or lemon juice, if desired.
Let the aioli sit for at least 1 hour before serving to allow the flavors to meld.
Transfer the aioli to a serving dish and top with fresh cracked black pepper.