In a large pot, melt 2 tablespoons of unsalted butter or olive oil over medium heat. Once melted, add 1 diced medium onion and 3 minced garlic cloves. Sauté for about 3-5 minutes until the onion is translucent and fragrant.

Stir in 2 diced medium carrots and 2 diced medium potatoes. Cook for an additional 5 minutes, stirring occasionally, until the vegetables begin to soften.

Sprinkle 1/4 cup of all-purpose flour over the sautéed vegetables, stirring well. Cook for about 2 minutes, allowing the flour to create a roux to thicken the soup.

Gradually whisk in 6 cups of vegetable broth, ensuring no lumps. Add 1 teaspoon of dried thyme, 1/2 teaspoon of dried rosemary, and salt and pepper to taste. Bring to a simmer and cook for 15 minutes.

Add 2 cups of broccoli florets, 1 cup of frozen peas, and 1 cup of corn kernels. Stir gently and cook for another 5-7 minutes until all vegetables are tender.

Lower the heat and pour in 1 cup of whole milk or half-and-half, along with 1/2 cup of heavy cream if desired. Mix until combined, adjusting seasoning as needed.

Ladle the soup into bowls and sprinkle fresh parsley on top. Serve hot with crusty bread or a side salad.