Heat oil in a skillet over medium heat. Add ham and cook until browned and crispy, stirring frequently, about 7 minutes.
Beat eggs with 1 teaspoon chives, salt, and pepper in a small bowl and add to skillet, tilting skillet to distribute egg evenly on bottom of skillet. Cook until omelette has set at the edges. Sprinkle Cheddar cheese on top and fold omelette in half. Flip and cook until egg is entirely set and cheese inside has melted, 30 seconds to 1 minute. Slide onto a plate and sprinkle with remaining 1 teaspoon chives.