Heat the olive oil in a large skillet over medium heat. Add the sliced onions and cook slowly, stirring frequently, until they become golden brown and caramelized, about 30-40 minutes. Sprinkle with a splash of water if needed to prevent sticking and to help with even caramelization.
Once the onions are deeply caramelized, stir in the balsamic vinegar. Cook for another 2-3 minutes, allowing the vinegar to meld into the onions and intensify their sweetness.
Transfer the caramelized onions to a food processor or blender, and pulse until finely chopped but not completely pureed; you want some texture remaining.
In a mixing bowl, combine the softened cream cheese, sour cream, and mayonnaise. Mix until smooth and creamy, with no lumps remaining.
Add the chopped caramelized onions to the creamy mixture. Mix thoroughly to incorporate the onions evenly throughout the dip.
Season the dip with salt and freshly ground black pepper. Taste and adjust the seasoning as desired.
Transfer the dip to a serving bowl, cover, and refrigerate for at least 30 minutes to allow flavors to meld and the dip to chill.
Serve the French onion dip with crispy chips or sliced vegetables, and enjoy the rich, savory flavors with a smooth, velvety texture.