Peel and dice the onion, carrots, and potatoes. (Dice the potatoes into 1/2 ich cubes. The smaller that you dice the potatoes the faster they will cook.)
Add the diced onion and minced garlic to 2 tablespoons of oil then sprinkle with 2 teaspoons of salt and sauté for 5 minutes over low heat.
Add the vegetable broth, water, carrots, and potatoes, then turn the heat to medium-high and bring the soup to a slow boil.
Simmer the potatoes and carrots for about 10 minutes until the potatoes are tender.
While the soup is simmering wash and chop the broccoli into small bite-sized florets. Then add the broccoli after the potatoes have simmered for 10 minutes.
Continue to simmer the soup for an additional 10 minutes until the broccoli is tender.
Make the cornstarch slurry by dissolving 1/3 cup of cornstarch in 1/2 cup cold water. (You can use a vegan roux to thicken the broth instead if you choose.)
Add the plant-based milk, cornstarch slurry, turmeric, pepper, and nutritional yeast, and simmer for 5 more minutes. Then add additional salt and fresh ground black pepper to taste.
Notes
If you don't have nutritional yeast, you can leave it out. It just provides a slightly cheesy flavor to the broth.
Be sure to add the broccoli after the potatoes have cooked a little so your broccoli doesn't get overcooked.
To make this soup cheesy, add a cup of vegan cheddar cheese to the pot of soup at the end of cooking and garnish with additional cheese when serving.