Begin by melting the butter and olive oil together in a large pot over medium heat, allowing it to become fragrant and bubbly—this takes about 1-2 minutes.

Add the finely diced yellow onion to the pot. Cook, stirring occasionally, until the onion is caramelized and golden brown, which should take about 5-7 minutes.

Stir in the minced garlic along with the smoked paprika, ground mustard, and cayenne pepper. Sauté for about 1 minute until the garlic is aromatic.

Next, add the remaining 4 tablespoons of butter with the all-purpose flour. Cook for 2 minutes until golden; this roux will help thicken your soup.

Gradually pour in the chicken or vegetable broth and half-and-half. Whisk everything together until you have a smooth and combined mixture.

Toss in a bay leaf if using, and let the soup simmer uncovered for 15 minutes. This will help meld the flavors beautifully.

Stir in the chopped broccoli and shredded carrots. Continue to simmer until the vegetables are tender, about 10 minutes.

Remove the pot from heat and gradually stir in the shredded extra-sharp cheddar cheese until completely melted, creating that creamy goodness.

Before serving, give your soup a taste and adjust the seasoning with salt and pepper as needed for the perfect flavor.