In a large pot, melt 2 tablespoons of unsalted butter or olive oil over medium heat. Once shimmering, add 1 diced medium onion and 2 minced garlic cloves, sautéing until the onion becomes translucent and fragrant, about 3-4 minutes.

Add 2 diced medium carrots and 2 diced medium potatoes to the pot, stirring well to combine. Cook these vegetables for about 5 minutes until they start to soften but still retain some crunch.

Sprinkle 2 tablespoons of all-purpose flour over the sautéed vegetables, stirring constantly for about 1-2 minutes until they are well-coated.

Gradually whisk in 4 cups of vegetable broth, stirring continuously to prevent any lumps from forming. Increase the heat slightly and bring the mixture to a gentle simmer, which should take about 3-4 minutes.

Stir in 1 teaspoon of dried thyme, 1/2 teaspoon of dried rosemary, and salt and pepper to taste. Lower the heat and allow the mixture to simmer uncovered for about 15 minutes.

Once the potatoes are tender, stir in 2 cups of broccoli florets, 1 cup of frozen peas, and 1 cup of corn kernels. Cook for an additional 5-7 minutes.

Reduce the heat to low, then stir in 1 cup of whole milk or half-and-half, along with 1/2 cup of heavy cream if desired. Simmer gently for about 3-5 minutes.

Taste and adjust the seasoning of your soup if necessary, adding more salt and pepper as desired. Ladle the hot soup into bowls and garnish with freshly chopped parsley.