Heat the oil over medium heat in a Dutch oven. Add the onion, carrot, and celery. Cook, stirring occasionally, until softened, 10 to 12 minutes, adding the garlic during the last minute.
2Add the flour and cook, stirring constantly, until evenly combined and vegetables are well coated, about 1 minute. Gradually stir in the chicken broth and bring to a simmer. Add the chicken, thyme, salt, and pepper, and return to a simmer over medium. Reduce the heat to medium-low and simmer until the chicken is cooked through, about 5 minutes.
3Stir in the gnocchi and cook, covered, stirring occasionally, under tender and fluffy, 4 to 5 minutes. Stir in the spinach and half-and-half. Cook, stirring occasionally, until well combined and spinach is tender, 1 to 2 minutes. Remove from heat. Season to taste with salt and pepper.