In a medium pot, melt 1 tablespoon of butter over medium heat. Add the finely diced yellow onion and sauté until it turns translucent, about 3-4 minutes. Next, add 2 cloves of minced garlic, stirring constantly for another 30 seconds until fragrant.

Add 2 tablespoons of butter to the pot and allow it to melt completely. Once melted, sprinkle in 2 tablespoons of all-purpose flour, stirring continuously for about 2 minutes to cook the flour, forming a roux.

Gradually whisk in 4 cups of vegetable stock and 1 cup of half and half, ensuring there are no lumps. Bring the mixture to a gentle simmer for approximately 15-20 minutes.

Once the base has thickened, add 4 cups of fresh broccoli florets and 1 cup of thinly sliced carrots into the pot. Cook for an additional 15 minutes.

Stir in 1 tablespoon of chicken bouillon, 1 teaspoon of smoked paprika, and salt and pepper to taste. Allow it to cook for another minute, then mix in 2 cups of freshly shredded sharp cheddar cheese until melted and creamy.