Heat the extra-virgin olive oil and butter in a large, deep skillet over medium to medium-high heat. Add the ground chicken. Brown the meat 3 to 4 minutes, push it off to the sides of the pan then add mushrooms, garlic, onions and peppers to the center of the pan. Cook the vegetables 5 minutes then mix them together with chicken and season with smoked paprika, salt and pepper. Stir in stock and tomato sauce and bring to a bubble. Reduce heat to a simmer and cook 5 minutes then stir in sour cream and turn off heat. Add parsley. Adjust seasonings, to your taste.
While chicken goulash cooks, bring a medium pot of water to a boil. Salt boiling water and cook gnocchi until they puff up like pillows and float, 5 minutes or to package suggestions. Drain gnocchi and add to chicken just after you stir in the sour cream.
Ladle goulash with gnocchi into shallow bowls and garnish with chives.