Heat a low large sauté pan (or braiser) with a tight-fitting lid over medium heat. Add the butter and oil and cook until the butter is melted, swirling the pan a few times to coat the bottom. Add the chopped onion, shallots and celery. Cook for 3-4 minutes until the onion is softened.
If using the wine, add it now, then increase the heat a bit and allow the wine to reduce for 3-4 minutes.
Add the flour, stirring frequently for 30-60 seconds to coat the veggies.
Slowly add the chicken broth, stirring to dissolve all of the flour then add the cream or half and half, carrots, Italian seasoning, bay leaves, rosemary and garlic salt. Bring mixture to a medium boil. Once the sauce starts to boil, cook UNCOVERED, for 10 minutes, stirring and scraping the sides of the pan occasionally.
Remove the bay leaves. Add the chopped chicken and stir to combine. You can add a bit more chicken broth at this point if the sauce seems a little thick. (It will get thicker as the dumplings cook.)
Add tablespoons of the dumpling mixture (see recipe below) to the simmering chicken mixture. (I like to use a small 1½-2 tablespoon cookie scoop but a spoon can also be used. Don’t make the dumplings too large, they will get significantly bigger as they cook.)
Cover the pot and simmer on low heat for 15 minutes without removing the cover. The dumplings should be fluffy and float to the top. You can cut one open to make sure it’s cooked. If needed cover the pot and cook for another 2-3 minutes. Don’t overcook the dumplings as they will fall apart.