Melt the butter in a small nonstick skillet over medium heat. Add the ham and bell pepper and cook, stirring occasionally, until the ham is slightly browned and the pepper is softened, 3 to 4 minutes.
Meanwhile, whisk together 2 of the eggs and some salt and pepper in a medium bowl. Stir in the Cheddar.
Push the ham and peppers to the outside of the skillet, creating a ring. Pour the whisked egg mixture into the middle of the ring and let cook until mostly set, 1 to 2 minutes. Crack the third egg into the middle of the skillet on top of the omelet. Cover and cook until the egg around the edges is set but the yolk in the middle is still runny, about 4 minutes. Slide out of the skillet onto a plate and serve immediately, sprinkled with chives.
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