Prepare the ingredients. Chop the fresh broccoli into small florets, dice the onion, mince the garlic, and shred the carrots. Grate the sharp cheddar from a block and cut the Velveeta into cubes.
Sauté the vegetables. Heat the cooking oil in a large pot over medium heat. Add the onion, garlic, and carrots. Cook, stirring occasionally, until the vegetables soften and become fragrant.
Make the roux. Add the butter and allow it to melt. Reduce the heat to medium if needed, then stir in the flour. Cook for approximately 30 seconds, stirring to combine it with the butter and vegetables.
Add the liquids. Slowly pour in the chicken broth, stirring to incorporate it into the roux. Add the whole milk next, continuing to stir until the mixture is smooth.
Cook the broccoli. Stir in the chopped broccoli and bring the soup to a gentle simmer. Reduce the heat to medium-low, cover, and cook for approximately 5 minutes, or until the broccoli is tender.
Melt the cheeses. Keep the heat low and add the cubed Velveeta and freshly shredded sharp cheddar. Stir until both cheeses have completely melted and the soup is smooth and creamy. Avoid boiling the soup.
Season and serve. Turn off the heat once the cheeses have melted. Taste carefully and season with salt and black pepper as needed. Serve warm.