In a large pot, heat the olive oil until shimmering. Add the onions and cook over moderately high heat, stirring occasionally, until golden brown, about 15 minutes.
Tie the parsley, bay leaves and rosemary together with kitchen twine. Add the herb bundle to the diced onions and cook over low heat, stirring a few times, until fragrant, about 3 minutes. Sprinkle the sugar over the onions and cook, without stirring, until the sugar melts, about 5 minutes. Increase the heat to high and cook, without stirring, until an amber-brown caramel forms, about 6 minutes. Stir in the white balsamic vinegar and simmer over low heat, stirring a few times, until the jam is thick, about 5 minutes. Discard the herb bundle. Season the jam with salt and let cool to warm.
The caramelized onion jam can be refrigerated for up to 5 days. Serve warm or at room temperature.