Peel potatoes, slice in half lengthwise then slice into ~1/2-inch planks. Place the fries in bowl filled with cold water; allow them to soak in the water for at least 30 minutes; this makes the fries more crispy when cooked as it removes some of the starch.
While the fries soak, prepare the aioli: Add the egg, lemon juice, garlic, Dijon, olive oil and basil leaves to the bowl of a food processor or the pitcher of an immersion blender. Process until the mixture emulsifies, becoming thick and creamy. Season to taste with salt and pepper. Transfer mixture to a small bowl and refrigerate until ready for use, to allow the flavors to meld.
Drain and pat dry the fries. In a large bowl, toss the fries with the olive oil, balsamic, salt and pepper.
Preheat an air fryer to 400°F. Working in batches, lay out the fries in a single layer in the fryer. Cook for about 8 minutes then remove basket and gently toss/flip the fries; cook another 4 to 6 minutes or until they’ve browned and crisped. Remove from the basket and sprinkle with salt. Repeat with remaining fries.
Serve warm fries with the basil aioli.