Heat the olive oil in a large pot over medium heat. Add the diced onion and sauté until translucent, about 5 minutes.
Stir in the minced garlic and cook for an additional minute until fragrant.
Add the diced potato and broccoli florets to the pot, followed by the vegetable broth. Bring the mixture to a boil, then reduce heat and simmer until the broccoli and potatoes are tender, about 15-20 minutes.
Remove the pot from heat and use an immersion blender to blend the soup until smooth and creamy. If you don’t have an immersion blender, you can carefully transfer the soup to a countertop blender and blend in batches.
Return the blended soup to low heat. Stir in the heavy cream and shredded cheddar cheese, mixing until the cheese is melted and the soup is velvety.
Season with salt, black pepper, and smoked paprika (if using) to taste, adjusting seasoning as desired.
Serve hot, garnished with fresh parsley for a pop of color.