For the cakes: Place all ingredients, except for the oil, into a mixing bowl and stir together until well combined. Season with salt and allow the mixture to sit for 5 minutes.
Pour oil into a large sauté pan and place over medium heat.
Form ¼ cup patties with the quinoa mixture and place in the heated sauté pan. (this will have to be done in batches)
Cook quinoa cakes for about 5 minutes on each side. Repeat with the remaining patties until all of the cakes have been cooked. Set aside.
For the aioli: Place all ingredients into a food processor, season with salt and pepper and blend until smooth. Pour into a small bowl and serve alongside the warm quinoa cakes.