Bring a large pot of water to a boil. Add 1 tablespoon salt and pasta. Carefully add eggs and return to a boil. Cook until pasta is tender with a bite, about 9 minutes. Drain. Remove eggs and add to an ice bath. Rinse pasta with cold water and drain well. Add pasta to a large bowl.

Peel eggs; separate yolks and whites. Roughly chop egg whites and add to pasta bowl. Place yolks in a medium bowl and set aside.

Reserve 2 tablespoons green onions for garnish and add remaining green onions to pasta bowl. Add in celery, carrots, and pickles.

Mash egg yolks in the medium bowl and add remaining salt, mayonnaise, sugar, vinegar, mustard, and pepper. Stir mixture until smooth and pour over pasta mixture. Toss until well combined.