Heat a skillet with 2 tablespoons of extra-virgin olive oil over medium heat and add the onions. Cook until the onions are golden brown and caramelized, for about 10 minutes. Season with salt and pepper, to taste. If the pan is getting dry, add some water. Set aside.
On high heat, add the remaining extra-virgin olive oil in another skillet and lay in the beef when the oil is hot. Sear both sides for about 30 seconds, turning only once.
Add the balsamic vinegar to the beef and deglaze the pan. Cook until the balsamic is reduced to a syrupy consistency.
Plate the beef tenderloins with the caramelized onions on top. Drizzle the beef with the balsamic reduction. Serve hot.