Bring potato, 1/2 tsp. salt, and water to cover to a boil in a large Dutch oven over medium heat, and cook 10 to 15 minutes or just until tender; drain.
Place eggs in a single layer in a stainless-steel saucepan. (Do not use nonstick.) Add water to a depth of 3 inches. Bring to a boil; cover, remove from heat, and let stand 15 minutes.
Meanwhile, chop green onions. Drain eggs, and return to pan. Fill pan with cold water and ice. Tap each egg firmly on the counter until cracks form all over the shell. Peel under cold running water. Chop eggs.
Stir together eggs, green onions, bell pepper, celery, and carrot in a large bowl; add potato, stirring gently to combine.
Stir together mayonnaise, pepper, and remaining 1 tsp. salt; add to potato mixture, stirring gently to coat. Season with salt to taste. Cover and chill 1 1/2 to 24 hours before serving. Garnish, if desired.