Begin by washing and cutting your broccoli into small florets. Set them aside.

In a large pot, melt the butter over medium heat. Add in the diced onions and carrots, cooking until softened, about 5 minutes.

Sprinkle the flour over the sautéed vegetables and stir to combine. Cook for 2-3 minutes to eliminate the raw flour taste while stirring continuously.

Gradually whisk in the vegetable broth, making sure to break up any lumps created by the roux. Bring the mixture to a gentle simmer.

Pour in the half-and-half and stir well. Allow this to warm gently, then add the broccoli florets, garlic powder, salt, and pepper. Cook until the broccoli is tender, about 10 minutes.

For an ultra-smooth soup, use an immersion blender to puree the soup to your desired consistency or leave it chunky if you prefer.

Lastly, remove from heat and stir in the cheddar cheese until melted and smooth. Taste and adjust the seasoning if necessary.

Ladle into bowls, top with extra cheese if you please, and serve hot!