In a large pot or Dutch oven, heat the olive oil over medium heat. Add the chopped yellow onion, diced carrots, and diced celery. Cook for 8-10 minutes, stirring occasionally, until the vegetables are softened. Add the minced garlic and cook for another minute until fragrant.

Push the vegetables to one side of the pot. Add the unsalted butter to the empty side and let it melt. Once melted, sprinkle in the all-purpose flour and whisk constantly for 1-2 minutes to create a smooth paste (roux).

Gradually whisk in the chicken broth, ensuring no lumps form. Bring the mixture to a gentle simmer, stirring frequently, until the soup begins to thicken slightly.

Reduce the heat to low. Stir in the milk, heavy cream, dried thyme, and dried rosemary. Season with salt and freshly ground black pepper to taste. Continue to heat gently, stirring occasionally, until the soup is heated through. Do not bring to a rolling boil after adding the dairy, as it can curdle.

Add the shredded cooked chicken and the potato gnocchi to the soup. Cook for 3-5 minutes, or according to gnocchi package directions, until the gnocchi are tender and float to the surface.

If using fresh spinach, stir it into the soup during the last minute of cooking, allowing it to wilt.

Taste the soup and adjust seasonings as needed.

Ladle the hot soup into bowls. Garnish generously with grated Parmesan cheese and fresh parsley or basil, if desired. Serve immediately with crusty bread for dipping.