Bring 4 1/4 cups water, 1 teaspoon salt and 1/2 teaspoon pepper to a boil in a large saucepan. Stir in the oats, reduce heat to medium and cook, stirring constantly, until the oats are soft and the oatmeal is slightly thickened, about 7 minutes. Stir in the ham, Gruyere and chives and remove the saucepan from the heat.
Heat the oil in a large nonstick skillet over medium-high heat. Carefully crack the eggs into the pan and cook until the whites are set and slightly crispy and the yolk is still runny, 2 to 3 minutes. Sprinkle with 1/4 teaspoon salt.
Transfer the oatmeal to individual bowls. Top each with an egg, sprinkle with additional fresh chives and serve.
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