Sauté the Aromatics: In a large pot, melt the butter over medium heat. Add the chopped onion and cook for 4–5 minutes until softened. Stir in the minced garlic and cook for an additional 1–2 minutes until fragrant.
Cook the Potatoes: Add the diced potatoes and broth to the pot. Bring the mixture to a boil, then reduce the heat and let it simmer for 10–15 minutes, or until the potatoes are tender.
Prepare the Broccoli: While the potatoes are cooking, blanch the broccoli florets in boiling water for 3–4 minutes. Drain and set aside.
Blend the Soup: Once the potatoes are soft, use an immersion blender to puree the soup until smooth and creamy. Alternatively, blend the soup in batches using a blender.
Incorporate Cream and Cheese: Stir in the heavy cream, shredded cheddar cheese, salt, black pepper, and dried thyme (if using). Mix until the cheese is fully melted, creating a creamy consistency.
Add the Broccoli: Gently fold the blanched broccoli into the soup. Let it simmer for another 3–5 minutes to heat everything through.
Taste and Serve: Adjust the seasoning as needed, then ladle the soup into bowls. Garnish with chopped fresh parsley and serve warm.