Preheat oven to 425 degrees F (220 degrees C). Lightly grease a baking sheet.
Place potatoes on prepared baking sheet. Bake potatoes in the preheated oven until tender, about 30 minutes.
Heat olive oil in a large soup pot over medium heat. Stir celery, carrots, onion, and garlic into the hot olive oil; season with salt and pepper. Cook and stir until vegetables soften, about 10 minutes.
Stir chicken broth and milk into vegetable mixture; bring to a boil. Add cauliflower to vegetables and chicken broth mixture. Reduce heat to medium and cook until cauliflower is tender, about 10 minutes.
Stir potatoes, chicken, and Parmesan cheese into vegetables and broth; simmer until chicken is heated and cheese is melted, 20 minutes.
