Place the chicken breasts in a medium saucepan and cover with water; add 1 teaspoon salt. If breasts are large, cut in half. Bring to a boil, then reduce heat, cover and simmer 15–20 minutes until cooked through. Drain, let cool, remove skin if using rotisserie, and dice the meat.
Peel and dice the carrots and potatoes into similar-size pieces. Put them in a medium pot, cover with water, add 1 teaspoon salt, and bring to a boil. Cook uncovered 10–12 minutes until tender but not mushy.
Drain the carrots and potatoes in a colander and rinse with cold water to stop cooking. Pat dry with paper towels to remove excess moisture.
In a large mixing bowl combine the diced chicken, cooked potatoes, cooked carrots, thawed peas, diced hard-boiled eggs, diced pickles, and finely diced onion.
Add the chopped fresh dill and mayonnaise. Stir gently until evenly combined, then season with 1/2 teaspoon salt and 1/4 teaspoon pepper, adjusting to taste.
Chill for at least 30 minutes before serving to let flavors meld, if desired. Serve as a side.