Heat olive oil in a saucepan. Season chicken pieces with salt and pepper; cook and stir in the hot oil until no longer pink in the middle, 7 to 10 minutes. Remove chicken to a bowl using a slotted spoon, retaining drippings in the pan.

Pour chicken broth into the saucepan. Bring broth to a boil while scraping the browned bits of food off of the bottom of the pan with a wooden spoon; continue cooking until the broth reduces in volume by about half, 7 to 10 minutes. Return cooked chicken to the saucepan. Stir pesto through the chicken mixture; remove from heat.

Bring a large pot of lightly salted water to a rolling boil. Cook gnocchi at a boil until they float to the top, about 3 minutes. Remove gnocchi from the water to a large bowl using a slotted spoon, retaining water in the pot.

Place the saucepan with the chicken and pesto over the boiling water; cook and stir over the boiling water until warmed completely, about 5 minutes. Pour chicken and pesto mixture over the gnocchi; add mozzarella and stir until evenly mixed.