Sauté the base veggies: Bring a large nonstick pot to medium-high heat on the stovetop. Add a splash of water to prevent sticking, then add the onion, celery, and carrot. Sauté for 4 minutes, stirring occasionally and adding small amounts of water as needed to prevent sticking. Add garlic to the pot and sauté another minute, until fragrant.
Boil the soup base: Add the veggie broth, potatoes, cashews, and bay leaf. Bring to a boil, then lower the heat to a simmer and cover with a lid. Simmer for 10-15 minutes, or until the potatoes are easily pierced with a fork. Using kitchen tongs, carefully remove and discard the bay leaf.
Blend the soup: Carefully transfer the contents to a high-powered blender. Add the dijon mustard, salt, paprika, and nutritional yeast. Blend until very smooth.
Steam the broccoli: Place the broccoli florets in a steamer basket and set over a pot with an inch of water in it. Bring the water to a simmer, and steam the broccoli for 5 minutes, or until tender.
Add the broccoli: Transfer half of the steamed broccoli to the blender and pulse a few times to break it into smaller pieces. Pour the blended soup back into the large pot, then stir in the remaining steamed broccoli. Bring the soup back to a simmer and simmer for 5 minutes. Finishing touches: Stir in the lemon juice and apple cider vinegar. If using the optional vegan cheddar, add it now and stir until melted and incorporated. Taste and adjust seasonings as needed.Serve with additional shredded vegan cheddar, if desired. Enjoy!