Heat oil over medium heat in a Dutch oven. Add onions, carrots, and celery. Cook, stirring occasionally, until vegetables are softened—10 to 12 minutes. Add garlic during the last minute.
Add flour and cook. Stir constantly until evenly combined and vegetables are well coated, about 1 minute. Gradually stir in chicken broth and bring to a simmer.
Add chicken, thyme, salt, and pepper, and return to a simmer over medium. Reduce heat to medium-low and simmer until chicken is cooked through.
Stir in gnocchi and cook, covered but stirring occasionally, until tender and fluffy.
Add spinach and half-and-half. Cook, stirring occasionally, until ingredients are well-combined and spinach is tender.
Remove from heat. Season to taste with salt and pepper and serve.