Heat olive oil in a pot over medium heat. Add onion and carrots and sauté until the onions are translucent, soft, and fragrant, about 7-8 minutes.
Add garlic and sauté for 1 minute, until fragrant and slightly golden, being careful not to burn the garlic.
Add vegetable broth, chopped broccoli stems and turmeric powder. Bring to a boil, then lower the heat to low, place the lid on the pot and simmer for about 15-20 minutes or until the vegetables are really tender. Let cool slightly.
While waiting, add water into a saucepan and bring to a boil. Add the chopped broccoli florets and cook for just 3-5 minutes, just until tender but still bright green in color. Drain and set aside.
Once the soup has finished simmering, add nutritional yeast and Dijon mustard. Then, use an immersion blender to carefully blend until creamy. Alternatively, carefully transfer the soup to a blender and blend. Work in batches if needed and be careful as the liquid is very hot. The soup should be thick. If you like your soup thinner, add a splash of water of vegetable broth to adjust for consistency.
Add your broccoli florets back into the soup and stir to combine.
Season to taste with salt and pepper, adding more nutritional yeast for more cheesy flavor and mustard for the tang.