Add 1 Tbsp butter to dutch oven and heat over MED heat. Add chopped onion and saute until onions are soft and translucent, about 2-3 minutes. Add 1/4 cup butter to dutch oven, still over MED heat. When melted, add flour and whisk together. Cook mixture about 1 minute, whisking often.

Slowly pour in about a cup of chicken broth, whisking as you pour. Continue with remaining chicken broth and half and half, pouring slowly and whisking as your pour.

Let broth/half and half mixture cook for a minute or two, whisking often, making sure no flour lumps remain.

Add broccoli and carrots and stir to combine. Add paprika, garlic powder, dry mustard, salt and pepper, stir. Reduce heat to LOW/MED LOW and simmer about 10-15 minutes or until thickened and broccoli/carrots are fork tender. If soup gets too thick for your liking, add a splash of chicken broth or half and half to loosen it up.

Blend soup until desired texture is reached, using either immersion blender or by transferring soup to traditional blender.

Add grated cheddar cheese a handful at a time, stirring to melt the cheese into the soup after each handful. Taste soup and season with additional salt and pepper if needed.

Serve hot with additional black pepper and shredded cheese on top if desired.