In a heavy pan (like a dutch oven), combine butter and olive oil over medium heat.
Once the butter is melted, add carrots, celery, and onions, stirring to combine. Cook for 10 minutes.
Add garlic. If using raw diced chicken, add here. Cook for 5-10 more minutes.
Sprinkle flour over the veg and chicken, and gently stir to combine.
Slowly add the half and half, stirring to combine.
Add the chicken broth and parsley, tarragon, thyme, smoked paprika, salt, and pepper.
Crank the heat up to medium-high, and bring the mixture to a gentle boil. Stirring constantly, cook on medium-high for 2-4 minutes.
Once the mixture seems to thicken up and looks a bit cloudy, add the gnocchi, and reduce heat to medium-low. You'll need to keep an eye on it and stir occasionally. If using leftover cooked chicken, add that now.
The soup is ready when the (raw) chicken is cooked through, and the gnocchi are floating near the top.
Half and half seems to be an American ingredient. It is basically a mixture of heavy cream and milk. If you can't find it in stores near you, simply use 1 cup heavy cream, and 1 cup of your preferred milk.Add a pinch (~1/8 tsp) of nutmeg to really bring out the flavor of the soup.