For the caramelized onions: Heat a nonstick skillet over medium heat. Add the olive oil and onions, separating them, and sprinkle with the salt. Saute for 10 minutes, then turn down the heat to medium-low and cook until the onions are deep amber in color, another 35 to 40 minutes.
Meanwhile, for the fig jam: Combine 1/4 cup water, the figs, sugar and lemon zest in a saucepan and simmer over low heat until thick, 45 to 60 minutes. For the eggs: Heat a medium nonstick skillet over medium heat. Add 1 teaspoon olive oil and use a paper towel to grease the pan and remove the excess oil. Crack 2 egg whites into a bowl and beat with a whisk. Pour the whites into the skillet and add 1/4 teaspoon piment d'espelette and some black pepper. Cook until opaque, 1 to 3 minutes. Flip and cook for another minute. Transfer to a plate. Repeat with the remaining eggs to make 3 more omelets.
For the salad: Combine the olive oil, vinegar and maple syrup in a small jar with a tight-fitting lid. Seal the jar and shake vigorously to blend. Toss the arugula and blueberries with the dressing. Toast the bread. Spread 1 tablespoon jam on each of 4 slices of toast and add 1 tablespoon caramelized onions. Top with the egg whites and cover with the remaining slices of toast. Serve with the salad.