In a dutch oven melt butter over medium-high heat.
Add in onions to dutch oven and saute until soft and translucent, about 2-3 minutes.
Add in flour while whisking to form a roux, cook for about 1 minute whisking often. Continue to whisk and pour in about half of the broth.
Add in remaining broth, as well as the half and half, slowly whisking as you pour the broth and half and half into dutch oven.
Cook for about 2-3 minutes, whisking often to get out all the lumps, add in paprika, garlic powder, dry mustard, salt and pepper. Whisk to combine.
Remove from heat and transfer to a blender or food processor, this is optional, blend until smooth.
Return the soup base to the dutch oven and add in broccoli florets and diced carrots.
Reduce heat to medium-low and simmer for 15-20 minutes or until soup has thickened and the vegetables are fork tender, stir occasionally through cooking time.
Add shredded cheddar cheese a handful at a time, stirring to incorporate after each handful.
Season with extra salt and pepper as need and serve immediately.