Start by preparing all your ingredients. Ensure that the vegetables are finely chopped to ensure even cooking, and the potatoes are peeled and cubed to similar sizes for uniformity.
In a large, heavy-bottomed pot or Dutch oven, combine the water, chicken bouillon cubes, finely chopped onion, celery, carrots, broccoli, and cubed potatoes. Stir in the minced garlic, dried thyme, bay leaf, and a pinch of white pepper.
Place the pot over medium-high heat and bring the mixture to a boil. Once boiling, reduce the heat to medium and let it simmer. Cook until the potatoes are fork-tender, approximately 10-12 minutes.
Once the potatoes are tender, reduce the heat to low. Stir in the unsalted butter, allowing it to melt seamlessly into the soup. Gradually pour in the half and half, mixing it in to maintain a smooth consistency. Add the grated cheddar cheese and cheese whiz (or nacho cheese), and stir until all the cheeses melt and the soup becomes creamy.
Allow the soup to simmer gently on low heat for another 25-30 minutes. Ensure to stir occasionally to prevent the bottom from scorching. During this phase, the flavors meld, and the soup thickens slightly.
Remove the bay leaf and discard. Taste the soup and adjust the seasonings if necessary.
Ladle the soup into warm bowls. Garnish with finely chopped green onions and, if desired, a sprinkle of fresh dill or parsley. Serve alongside a fresh dinner roll or crusty bread for a complete, comforting meal.