Bring a large pot of salted water to a boil. Add eggs, reduce heat to medium, and simmer 6 minutes. Remove eggs; run under cold water to stop cooking, then peel.

Return water in pot to a boil. Add gnocchi and cook until they float, 1 to 3 minutes. Add peas; return to a boil, then stir in arugula. Drain, reserving 3/4 cup pasta water.

Melt butter in pot over medium-high heat. Add onion; cook until translucent, about 3 minutes. Add cream, cheese, and 1/4 cup reserved pasta water and simmer, stirring, 1 minute. Add gnocchi mixture; toss to combine, then simmer 30 seconds more. Add additional pasta water, 1 tablespoon at a time, as needed to create a creamy sauce. Season with salt and pepper and serve with halved eggs, sprinkled with more cheese and pepper.