Preheat the oven to 425 degrees. Toss the carrots, potatoes, and onion with olive oil, half the minced garlic, thyme, rosemary, salt, and pepper on a baking sheet. Roast for 20 minutes until the edges start to brown.
Add the roasted vegetables to the slow cooker along with the broccoli florets, remaining garlic, and chicken broth. Stir to combine.
Cover and cook on low for 4 hours or high for 2 hours until the broccoli is tender.
Stir in the shredded cheddar cheese and heavy cream. Continue cooking on low for 15 more minutes until the cheese melts fully. Taste and adjust salt and pepper if needed.
Ladle into bowls and serve warm.