Add 2 tablespoons of the olive oil to a heavy-bottomed pot or Dutch oven over medium-high heat. Pat the chicken dry with paper towel, season with S&P. Add the chicken to the pot, working in batches as needed. Cook for 4-5 minutes per side, until deeply golden brown. Transfer to a plate and set aside. The chicken will not be cooked through at this point – that’s fine!
Reduce the heat under the pot to medium. Add in the carrot, celery, onion & potato, seasoning with 1/2 teaspoon kosher salt. Cook, stirring occasionally, until softened, about 5 minutes. Add in the garlic & Italian seasoning, stirring to combine, and cook 1-2 minutes, until fragrant.
Dust flour over the softened veggies, stirring to coat. Cook for 1-2 minutes, stirring frequently, until deeply browned. While scraping the bottom of the pot with a wooden spoon, slowly pour in the chicken stock. Add the bay leaves, parmesan rind, & poultry herbs (tied together with twine or finely chopped.) Bring to a boil, then reduce to a simmer. Add the browned chicken into the pot. Cover & simmer for 20 minutes.
Meanwhile, as the soup simmers, prep the DeLallo Gnocchi Kit. Follow Step 1 of package directions to prepare the gnocchi “dumpling” dough – Combine package contents with water in a medium bowl. Stir with a fork until dough begins to form. Knead by hand until dough no longer sticks. Set aside.
Remove the chicken from the pot & carefully shred. Add the shredded chicken back into the pot along with the half & half, stirring to combine. Bring the soup up to a simmer. Using 2 spoons, freeform the gnocchi dough into tablespoon-sized dumplings, dropping the dumplings into the pot one by one. Once all the dumplings have been added to the pot, cover & simmer for 5 minutes, until the gnocchi is cooked through. Taste and season with additional salt or ground black pepper as needed.