Step 1: Start the Base – Melt the butter in a large pot over medium heat; add onions if using and cook about 3 minutes until soft and translucent.
Step 2: Add the Liquids and Veggies – Pour in chicken broth and half and half, stir well, then add broccoli, carrots, and celery; bring to a gentle boil, then reduce heat and simmer until broccoli is tender.
Step 3: Thicken the Soup – Mix cornstarch with milk until smooth, then slowly pour into the simmering soup while stirring; continue stirring for a few minutes until it thickens.
Step 4: Season the Flavor – Add garlic powder, black pepper, thyme, salt, and optional pinch of nutmeg; taste and adjust seasoning as needed.
Step 5: Melt in the Cheese – Lower heat to low, add Velveeta cubes and stir until melted; then add shredded cheddar and mix until smooth and creamy.
Step 6: Blend or Serve – Blend lightly for smoother texture or leave chunky; ladle into bowls, sprinkle extra cheese if desired, and serve hot.