Start by cleaning and prepping your fresh vegetables. Wash the broccoli florets under cold water to get rid of dirt or chemicals. Slice the carrots into thin strips so they cook evenly. Peel the onion and chop it finely. Smash the garlic cloves to add extra flavor.

In a large pot or Dutch oven, heat the olive oil over medium heat. Add the chopped onions and minced garlic, stirring until they become translucent and fragrant. This usually takes about 3-4 minutes. Next, add the broccoli and carrots to the pot, and stir to cover the vegetables with oil evenly.

Pour in your choice of chicken broth or vegetable broth and make sure it covers the vegetables adequately. Heat the mixture until it starts to boil gently, then reduce the heat and let it simmer until the vegetables are soft. When the vegetables are ready, mix in the milk and heavy cream to make the soup creamy.

In a separate small bowl, whisk the all-purpose flour with a little bit of water to form a smooth paste. Slowly add this mixture to the simmering soup while stirring all the time. This will gently thicken the soup. Let it cook a few more minutes until it's as thick as you like.

Carefully use a blender or immersion blender to puree the soup until it's smooth and creamy. You can blend the whole batch or leave small chunks of broccoli for added texture as you like. Make sure to blend in batches if using a traditional blender to avoid spillage.

Finally, stir in the grated Parmesan cheese allow it to melt, and blend seamlessly into your soup. Season with salt and pepper to taste. If you desire extra garnish, sprinkle some chopped parsley and shredded cheddar cheese on top of the served soup bowls for a good presentation.