Heat 1 teaspoon butter in a skillet over medium heat; cook chicken in the melted butter until no longer pink in the center and juices run clear, 5 to 7 minutes per side. Set aside to cool; chop chicken into bite-sized pieces.
Bring a large pot of lightly salted water to a boil. Cook seashell pasta in the boiling water, stirring occasionally until cooked through but firm to the bite, 10 to 12 minutes. Drain and return pasta to pot.
Place a steamer insert into a saucepan and fill with water to just below the bottom of the steamer. Bring water to a boil. Add asparagus and peas, cover, and steam until tender, 3 to 4 minutes. Remove vegetables from heat and keep covered.
Heat 1 teaspoon butter in a saucepan over medium heat; cook and stir onion in the melted butter until translucent, about 10 minutes. Stir cream into onion; add vegetables and chicken. Simmer until cream reduced slightly, 2 to 3 minutes. Fold Parmesan cheese into mixture. Remove from heat and cover saucepan until cheese is melted, about 2 minutes. Toss pasta in the cream sauce.