Melt some butter in a large deep pot on medium heat. Add chopped onion and garlic and fry them for 2-3 minutes.
Mix some flour and keep stirring continuously with a whisk for 2-3 minutes.
Add some vegetable stock, little at a time, and keep stirring with a whisk for about 5-7 minutes, until the mixture turns shiny. Add garlic powder, cayenne pepper, black pepper powder, and salt. Mix well.
Add broccoli florets and carrots, and mix well. Cover it with a lid and bring the soup to a simmer and cook for about 12 minutes.
Stir in some milk and cook for 2-3 minutes on low heat. Mix some cheddar cheese and cook on low heat for another 1-2 minutes until it melts.
Taste the seasoning and adjust as per taste. Serve hot.
Notes
Make sure milk is at room temperature or warm it up in a microwave for 30-40 seconds.
If you are on a gluten-free diet, add corn flour to the recipe as a thickening agent.
You can add corn starch as well, but make sure you are adding half the amount of corn starch than corn flour or all purpose flour in the recipe.
To make the soup more rich, you may add half cup heavy cream in the recipe.
You can use frozen broccoli as well in the recipe to save some time.