In a large Dutch oven or heavy-bottomed pot, heat the olive oil and melt the butter over medium heat.
Add the diced onion, celery, and matchstick carrots to the pot. Sauté for 5 to 6 minutes until the vegetables have softened and the onions are translucent.
Stir in the minced garlic and cook for approximately 60 seconds until fragrant.
Whisk in the all-purpose flour and cook for 2 minutes, stirring constantly to toast the flour and create a light roux base.
Gradually pour in the chicken broth followed by the half-and-half, whisking continuously to ensure a smooth texture without flour clumps.
Add the dried thyme, salt, and black pepper. Bring the liquid to a gentle simmer over medium-high heat.
Carefully add the potato gnocchi and the cooked chicken to the simmering broth. Reduce heat to medium-low and cook for 5 to 8 minutes, or until the gnocchi are tender and have risen to the surface.
Stir in the chopped spinach and cook for 1 to 2 minutes until the leaves are wilted and the soup is thoroughly heated.
Adjust seasoning with additional salt and pepper if required. Serve immediately.