Heat the butter and olive oil in a large pot over medium heat until melted and sizzling.

Sauté the finely diced onion until caramelized and golden brown, about 5-7 minutes.

Add minced garlic, smoked paprika, ground mustard, and cayenne pepper; sauté for about a minute.

Make the roux by adding the remaining 5 tablespoons of butter and the flour; cook for 2 minutes.

Gradually whisk in the chicken or vegetable broth and half-and-half until fully incorporated.

Simmer the soup for 15 minutes over low heat, adding a bay leaf, if desired.

Add the chopped broccoli and shredded carrots; simmer until tender, about 10 minutes.

Stir in the shredded cheddar cheese until melted and combined.

Taste and adjust seasoning with salt and pepper before serving.