Bring a large pot of water to a boil, add 1 tablespoon of salt and the pasta, then carefully add the eggs and return to a boil.
Cook until the pasta is tender but still has a bite, about 9 minutes.
Drain everything, remove the eggs, and transfer them to an ice bath.
Rinse the pasta with cold water and drain it well before adding it to a large bowl.
Peel the eggs, separate the yolks from the whites, roughly chop the whites, and add them to the pasta bowl.
Place the yolks in a medium bowl.
Reserve 2 tablespoons of green onions for garnish and add the remaining green onions, celery, carrots, and pickles to the pasta.
Mash the egg yolks, then stir in the remaining salt, mayonnaise, sugar, vinegar, mustard, and pepper until smooth.
Pour this mixture over the pasta and toss until everything is well combined.
Sprinkle with the reserved green onions and paprika, then serve immediately or refrigerate until ready.