Fresh farm eggs are the best when making aioli because firm golden yolks yield the best results. However, store-bought eggs will also work. If the eggs have been stored in the fridge, place them in a bowl of lukewarm water and allow them to sit for 15-minutes.
Add the whole egg and the egg yolk into a food processor and beat until they begin to thicken and emulsify (20 seconds).
Add the extra virgin olive oil to the grapeseed oil, and little by little, drip a third of the oil into the eggs while the food processor runs. The volume will increase; at this point, stop the food processor and scrape down the sides. Turn the food processor back on, and add the remaining oil mixture in a steady, continuous stream. Stop mixing.
Through the food processor porthole, add the Dijon mustard, Old Bay seasoning, and white wine vinegar start the food processor and continue to mix for 10 seconds.
Spoon the mixture into a bowl and fold in the chopped cucumbers.
Serve.