Begin by melting 1 tablespoon of butter in a large Dutch oven over medium heat. Once the butter has fully melted and is sizzling, add the chopped yellow onion. Sauté the onion for about 2-3 minutes, stirring occasionally, until it becomes soft and translucent.

Next, add ¼ cup of butter to the pot and allow it to melt completely. Once melted, whisk in the all-purpose flour to create a roux. Cook this mixture for about 1 minute, stirring constantly.

Gradually whisk in the chicken broth, one cup at a time, ensuring that no lumps remain. Continue to whisk until smooth before adding the remaining broth and the half and half. Bring the mixture to a gentle simmer over medium-high heat.

Now, stir in the freshly chopped broccoli and carrots. Season generously with paprika, garlic powder, dry mustard, kosher salt, and black pepper. Reduce heat to medium-low and let the soup simmer for about 10-15 minutes.

For the next step, blend the soup to your desired consistency. If you have an immersion blender, blend it right in the pot until smooth and creamy.

Once blended, stir in the cheddar cheese gradually over low heat, allowing it to melt after each addition. Adjust seasoning if needed.

Serve your creamy broccoli cheddar soup hot, garnishing with a sprinkle of black pepper and additional shredded cheese, if desired.