Fry the sage leaves: Heat a 6-to-8-inch nonstick skillet over high heat and add the oil. Have a tray lined with paper towels and a slotted spoon ready. When the oil begins to look thinner and spreads to the sides of the pan, turn off the heat and add the sage leaves. Stir them to coat with the oil and cook, stirring, until the sage pales slightly in color and gets slightly crisp, 45 seconds to 1 1/2 minutes. Use a slotted spoon to transfer the leaves to the paper towels. Season them immediately with salt and allow them to cool. Reserve the skillet.
Blend the eggs: In a medium bowl, whisk together the eggs, 1 teaspoon water and 1/4 teaspoon salt. Whisk only enough to integrate the eggs; you don't want to whip too much air into them or make them frothy.
Cook the omelet: Return the skillet to medium heat, remove the excess oil and add the butter, swirling it as it melts so it coats the pan. When the butter is melted (but not browned), reduce the heat to medium low and pour in the egg mixture. Use a fork to stir the eggs slightly, as if you were scrambling them. Then allow the eggs to cook, undisturbed, until they start setting in the middle. Sprinkle the cheese and sage leaves over them. Cook until the eggs look almost fully cooked and only slightly loose, 1 to 2 minutes.
Serve the omelet: Lift the handle of the pan, tilting the pan away from you and toward the heat. This tilting should cause the omelet to slide down in the pan a little. Using a heatproof spatula, fold the edge closest to you toward the center. Fold the other edge in toward the center and invert the pan over the center of a plate so the omelet lands seam-side down. Season with pepper.