Start by prepping your ingredients. Chop the vegetables. Cut the potatoes into approximately 1-inch cubes, prep the carrots by cutting into 1/2" thick pieces and chop the onion if using raw. For the broccoli, chop it into tiny florets (cut off as much stem as you can so the pieces are smaller).
Add your broth to a pot and throw in the onions, broccoli, carrots, garlic and bay leaf. Simmer on medium, covered for about 15-20 minutes, or until the veggies are tender.
While the veggies are cooking, start the potatoes boiling in a large pot. Drain the water when they are cooked through but not too soft.
Once the broth/veggies and potatoes are done, remove the bay leaf. Then pour the potatoes into the pot with the veggies and broth. Slowly stir in the half and half (you can also use whole milk or heavy cream).
Turn the burner onto a low heat, uncovered. Season with the remaining herbs and spices, adding a little extra or a little less of whatever you like. Personally, I love the allspice and nutmeg in this soup. It adds an unexpected richness to the overall flavor without being overwhelming.
Once the soup has warmed back up, add the cheese in small handfuls and stir until melted.
If the soup is thinner than you like after adding the cheese, you can add a thickening agent. (I prefer to use Ultra Gel, which is a modified food starch similar to cornstarch, but you can add it to hot or cold liquids without mixing it with water first and it won't separate.) Remove from heat and serve. Enjoy!