Bring 3 1/2 cups water, milk, and salt to a boil in a large saucepan. Gradually stir in grits. Reduce heat, and cook, uncovered, 35 minutes or until thick, stirring frequently.
Heat a medium nonstick skillet over medium-low heat. Combine eggs, milk, and salt in a medium bowl, stirring well with a whisk.
Add oil to pan; swirl to coat. Add egg mixture; cook 1 1/2 minutes or until desired degree of doneness, stirring constantly. Spoon 1 cup grits into each of 4 bowls; top each serving with 2 tablespoons shredded cheddar cheese, one-fourth of eggs, and 1 tablespoon sliced fresh chives.
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