Start by chopping the onion and mincing the garlic, setting them aside for sautéing.
Heat the olive oil in a large heavy-bottomed pot over medium heat until shimmering.
Add the chopped onion to the pot and sauté for about 5 minutes, stirring occasionally, until it becomes translucent and fragrant.
Stir in the minced garlic and cook for another minute until fragrant, filling the kitchen with a warm aroma.
Add the chopped potatoes and broccoli to the pot, stirring well to coat them with the aromatic base.
Pour in the vegetable broth, bringing the mixture to a boil, then reduce the heat to a gentle simmer.
Simmer uncovered for about 15-20 minutes, until the broccoli and potatoes are tender and the mixture is vibrant green.
Remove the pot from heat and use an immersion blender or transfer the soup in batches to a blender, blending until smooth and creamy.
Return the pureed soup to low heat and stir in the shredded vegan cheddar, nutritional yeast, and plant-based milk, warming gently until the cheese melts and the soup is velvety.
Season with salt and pepper to taste, stirring well to incorporate all flavors.
Once everything is heated through and the cheese is fully melted, serve the soup hot, garnished with fresh herbs if desired, and enjoy its comforting, cheesy goodness.
