Whisk together water, sage, pepper, salt, eggs, and egg whites in a mixing bowl.
Add 1 1/2 teaspoons of oil to a large skillet and heat over medium; swirl to coat.
Pour in about 1/4 of the egg mixture. Cook until edges have set, then flip and continue to cook until center is just about done.
Sprinkle on some cheese, fold in half, then remove from skillet.
Continue with remaining ingredients until 4 omelets have been cooked; add more oil to the skillet as needed.
Divide lettuce among 4 plates.
Add cashews and the juice of 4 oranges to a blender or food processor; blend until smooth then drizzle over the lettuce.
Top salads with orange wedges from the remaining 2 oranges (or leave on the side if desired), and serve alongside omelets.