In a large pot, heat the olive oil over medium heat. Add the chopped onion and sauté until translucent, about 5 minutes.
Stir in the minced garlic and cook for an additional minute until fragrant.
Add the diced potatoes to the pot and pour in the vegetable broth. Bring the mixture to a boil, then reduce heat and let it simmer for 10 minutes, or until the potatoes are fork-tender.
Add the broccoli florets to the pot and continue to simmer for another 5 minutes, until the broccoli is bright green and tender.
Carefully use an immersion blender to puree the soup until smooth. Alternatively, you can transfer the soup in batches to a blender.
Return the pureed soup to the pot over low heat. Stir in the heavy cream, shredded cheddar cheese, salt, black pepper, garlic powder, and onion powder. Mix until the cheese is completely melted and the soup is creamy.
Taste the soup and adjust the seasoning, if necessary. If it's too thick for your liking, add a bit more vegetable broth to reach your desired consistency.
Serve the soup hot, garnished with croutons and chopped chives if desired.
Notes
Optional garnishes include croutons and chopped chives.