Begin by washing the broccoli thoroughly and chopping it into small, bite-sized florets. Peel and dice the potatoes into uniform pieces, and slice the carrots thinly. Set aside.

In a large pot, heat 2 tablespoons of olive oil over medium heat. Add the diced onion, and sauté for about 5 minutes until it turns translucent and fragrant.

Stir in the minced garlic and continue to sauté for another minute, being careful not to let it burn.

Next, add the diced potatoes and sliced carrots into the pot, stirring everything together. Cook for about 5 minutes.

Pour in 4 cups of vegetable broth (or water) and bring to a boil over high heat. Carefully add the chopped broccoli, season with salt and pepper, then reduce to a simmer.

Allow the soup to simmer gently for 15-20 minutes until all vegetables are tender.

Remove the pot from heat and blend the soup until smooth and creamy.

Return the blended soup to the pot, adjusting thickness with more broth or water as desired. Taste and adjust seasoning if necessary.

Ladle the soup into bowls and garnish with Parmesan cheese, crushed red pepper flakes, and fresh parsley if desired. Serve hot.