- In a large pot over medium heat, melt butter with oil. Add carrots, celery, and onion and sauté for five minutes. Add garlic and sauté 30 seconds longer.
- Stir in thyme, chicken broth, potatoes, fennel, and season with salt and pepper to taste. Bring to a boil over medium-high heat, then reduce heat to medium, cover with lid and cook 15 minutes.
- Add broccoli and cook 5 minutes longer or until veggies are tender.
- Add half and half and stir.
- Remove from heat and add in cheddar cheese until melted.