Heat 4 tablespoons of the oil in a large skillet over high heat until it trembles, becomes aromatic, and easily coats the bottom of the pan. Add the onions and turn the heat to very low. Sprinkle with salt. Cook the onions uncovered, stirring occasionally, until they are the color of a polished mahogany table, about 1 hour. They will shrink dramatically. Transfer the onions to a bowl and let them cool. Add balsamic vinegar drop by drop until the flavor of the onions is complex but not vinegary. Sprinkle with fleur de sel if needed. Brush the bread with the remaining 2 tablespoons oil. Put a generous pile of onions on each slice of bread. Add a few grindings of pepper and serve.