Heat the oil in a large nonstick skillet over medium heat. Add the onion and ¼ teaspoon each salt and pepper and cook, stirring often, until golden, 15 to 18 minutes. Transfer to a plate; reserve the skillet.
Add the eggs to the skillet and cook, covered, 2 to 4 minutes for slightly runny yolks. Season with ¼ teaspoon each salt and pepper.
Divide the onion, fig jam, frisée leaves, and eggs among the brioche rolls.