In a large 5 to 6-quart Dutch oven or heavy soup pot, melt butter with olive oil over medium heat. Add the diced onion, carrots, and celery. Sauté for 5 to 6 minutes until the vegetables soften and become fragrant. Stir in minced garlic and cook for 1 minute until aromatic.

Sprinkle the flour evenly over the sautéed vegetables. Stir continuously with a wooden spoon for 1 to 2 minutes to cook off raw flour starch and create a rich paste coating the vegetables.

Slowly pour in the chicken broth in a steady stream while whisking continuously to dissolve the roux without lumps. Whisk in heavy cream, dried thyme, ground nutmeg, kosher salt, and black pepper. Bring to a gentle boil, then lower the heat to medium-low and simmer for 5 minutes until lightly thickened.

Add the shredded chicken breast and potato gnocchi into the simmering pot. Cook uncovered for 4 to 5 minutes, stirring occasionally, until the gnocchi float to the surface and become tender, pillowy, and cooked through.

Stir in the chopped baby spinach and grated Parmigiano-Reggiano cheese. Simmer for 1 to 2 minutes just until the spinach wilts into vibrant emerald ribbons and the cheese melts smoothly into the velvety broth. Remove from heat, ladle into warm soup bowls, and serve immediately with warm garlic breadsticks.