In a large nonstick pot, heat the butter and olive oil over medium-high heat. Add in the onion, carrot, celery, and garlic, and season with some salt.

Cook until vegetables have softened, about 5-6 minutes. Add in most of the broccoli, setting aside some florets to add in later.

Continue cooking the vegetables for 5-6 more minutes, until all veggies are soft. Add in just enough water to cover all the vegetables, and mix in the spices. Cover the pot, lower the heat to a simmer, and cook for 30-40 minutes.

When all the vegetables are cooked thoroughly and very soft, and most of the water has been absorbed, remove the soup from heat and let mixture cool off slightly.

Add veggies and any residual water to a blender, add in the half & half, and blend until smooth and creamy.

Return the soup to the pot, and stir in the cheese and the broccoli florets you had set aside before. Taste the soup, and add more seasoning as desired.

Let it cook until the latest broccoli has cooked through, 10-15 minutes, then serve!