Step 1: Place a large soup pot over medium heat. Add the olive oil and butter. When the butter is melted and slightly foamy, add the diced onion, celery, and carrots. Cook, stirring occasionally, for five to seven minutes, until the vegetables have softened and the onion looks translucent.
Step 2: Stir in the minced garlic, dried thyme, dried oregano, kosher salt, and black pepper. Cook for about one minute, stirring constantly, until the garlic is fragrant. Avoid browning the garlic.
Step 3: Sprinkle the flour evenly over the sautéed vegetables. Stir well so the flour coats everything and no dry spots remain. Cook for one to two minutes, stirring often, to remove the raw flour taste.
Step 4: Slowly pour in the chicken broth while stirring and scraping the bottom of the pot. Start with a small amount of broth and stir until the mixture looks smooth and thick, then continue adding broth in portions until all four cups have been added and the mixture is combined with no visible lumps.
Step 5: Reduce the heat to medium low. Gradually stir in the half and half. Allow the soup base to heat until it is steaming but not boiling, stirring often. Keeping the temperature gentle helps the dairy stay smooth.
Step 6: Add the chicken breasts to the pot, making sure they are mostly submerged in the liquid. Adjust the heat so the soup stays at a very gentle simmer. Cook for twelve to eighteen minutes, turning the chicken once or twice, until the chicken is cooked through and reaches at least one hundred sixty five degrees Fahrenheit in the thickest part.
Step 7: Use tongs to transfer the cooked chicken to a cutting board or plate. Let it rest for a few minutes, then shred it into bite size pieces with two forks. While the chicken rests, keep the soup at a low simmer.
Step 8: Increase the heat under the pot to bring the soup to a gentle boil. Add the potato gnocchi and cook according to the package directions, usually three to five minutes, until the gnocchi float to the surface and are tender.
Step 9: Lower the heat back to low. Stir the shredded chicken, chopped spinach, and grated Parmesan cheese into the pot. Simmer gently for two to three minutes, just until the spinach is wilted and the cheese has melted into the broth. Taste and add more salt and pepper if needed.
Step 10: Ladle the chicken gnocchi soup into bowls. Garnish with extra Parmesan cheese and chopped fresh parsley if you like. Serve hot with crusty bread or a simple side salad.