Sauté the onion. Heat the olive oil in a large pot over medium heat. Add the onion and cook for 3 to 4 minutes, until it begins to soften.
Add the garlic. Stir in the garlic and cook for about 30 seconds, just until fragrant. Don’t let it brown.
Build the soup base. Add the potato, carrots, cashews, vegetable broth, nutritional yeast, smoked paprika, and salt. Stir well and bring to a boil.
Simmer until tender. Reduce the heat, cover, and gently simmer for about 15 minutes. The potato and carrots are ready when you can easily pierce them with a fork.
Blend the base. Carefully transfer the mixture to a countertop blender and blend until completely smooth and creamy. Work in batches if necessary when blending the hot soup.
Adjust the consistency. Return the blended soup to the pot. Stir in water a little at a time, only as needed, until it reaches a thick but easily spoonable consistency.
Add the broccoli and cheddar. Stir in the chopped broccoli, vegan cheddar, and Dijon mustard. Gently simmer for about 10 minutes, stirring occasionally, until the broccoli is tender and the cheese has melted into the soup.
Finish the soup. Stir in the fresh lemon juice. Taste and add more salt if needed.
Serve. Ladle into bowls and serve hot.