In a large pot over medium heat, heat 1 tablespoon of butter and 1 tablespoon of olive oil. Add the diced onion and cook until golden brown, about 3–5 minutes.

Add minced garlic, paprika, ground mustard, cayenne pepper, and salt and pepper to taste. Stir constantly for about 45 seconds to 1 minute until fragrant.

Add the remaining 6 tablespoons of butter and let melt completely.

Whisk in the flour and cook for about 2 minutes until well combined.

Gradually whisk in the chicken broth, then slowly add the half-and-half while whisking. Reduce heat to medium-low and add the bay leaf.

Let it simmer for about 15 minutes, whisking occasionally.

After 15 minutes, add the chopped broccoli and shredded carrots, and continue to cook on low heat for an additional 10–15 minutes.

When the broccoli is tender but still bright green, gradually stir in the shredded cheddar cheese a handful at a time, preventing clumping.

Taste and adjust seasoning with salt if necessary.

Serve hot in bread bowls or regular bowls, garnished with herbs or additional cheese if desired.