Broccoli: Peel off the tough and fibrous outer layers of the stalks. Cut the broccoli heads from the stalks.
For the stems, I cut the stems in half horizontally before slicing them into smaller half-circle pieces. Set aside.
For the heads of broccoli, separate them into florets. Roughly chop the flowers into small pieces. Set aside.
Potatoes: Peel the potatoes and then cut them into 1- to 2-inch (2.5 to 5 cm) pieces.
Heat a large Dutch oven or pot over medium high. Once hot, melt the vegan butter. Add onion and garlic. Sauté until onion is translucent, about 2 minutes.
Add the diced potatoes and broccoli stalks to the pot. If you want extra flavor, you can choose to brown and char the broccoli a little.
Pour in the vegetable broth. Add miso paste if using. Season with salt and pepper to taste. Cover the pot and let it boil.
Leave the potatoes and broccoli stems to cook for 8-10 minutes until fork tender.
Once tendered. Allow to cook for a few minutes before transferring to a blender. Blend until smooth and creamy. You can use an immersion blender.
Return the soup to the pot. Add in oat milk, more salt to taste (if needed), nutritional yeast (if using), and vegan cheddar slices.
Mix in the chopped broccoli florets. Let the broccoli florets cook for 4-5 minutes or until tender to your liking.
Serve the soup hot and then top with more shredded cheddar cheese. Enjoy with bread of your choice.