Bring a large pot of water to a boil over high heat. Add potatoes and cook until fork tender, about 20 to 25 minutes. Drain water and return potatoes to pot.
In a food processor, pulse egg yolk, garlic, and white wine vinegar. Slowly drizzle in oils until aioli forms. {You can hand whisk the aioli, but it takes a while for it to really come together the way it should.}
Toss warm potatoes in 2 tablespoons aioli. Let potatoes rest for 20 minutes to really take on the aioli flavor. Serve on a platter with remaining aioli.