Start by melting the butter in a large pot or Dutch oven over medium heat. Once the butter bubbles, add the diced onions. Sauté the onions until they become translucent and fragrant, about five minutes.

Add the minced garlic to the pot. Stir constantly for about one minute to avoid burning the garlic.

Sprinkle the flour over the butter and onion mixture. Whisk continuously for two minutes to cook out the raw flour taste.

Slowly pour in the chicken stock while whisking to eliminate lumps. Gradually add the half and half, whisking until smooth.

Bring the mixture to a gentle simmer. Add the broccoli florets and grated carrots.

Reduce the heat to low, cover the pot, and let it simmer for about fifteen to twenty minutes until the broccoli is tender.

Turn off the heat and stir in the shredded cheddar cheese one handful at a time until melted.

Season with salt, black pepper, and smoked paprika. Serve immediately.