In a large pot over medium heat, melt 2 tablespoons of unsalted butter or olive oil until shimmering. Add one chopped medium onion and two minced garlic cloves, sautéing for about 2–3 minutes until the onion is translucent and fragrant.

Next, toss in two diced medium carrots and two diced medium potatoes to the pot. Sauté these vegetables for an additional 5 minutes, stirring frequently, until they start to soften and become slightly golden.

Sprinkle in 1/4 cup of all-purpose flour over the sautéed vegetables, stirring until they are well coated. Cook this mixture for about 1-2 minutes, allowing the flour to absorb flavors and form a roux.

Gradually whisk in 4 cups of vegetable broth, ensuring no lumps remain. Continue to stir until the mixture is smooth and begins to thicken slightly. Bring the soup to a gentle simmer over medium-low heat.

Add 1 teaspoon of dried thyme, 1/2 teaspoon of dried rosemary, along with salt and pepper to taste. Increase the heat slightly to bring the pot to a simmer and cook uncovered for about 15 minutes, until the potatoes are fork-tender.

Now, introduce 1 cup of broccoli florets, 1 cup of frozen peas, and 1 cup of corn kernels to the pot. Allow the soup to simmer for an additional 5-7 minutes, until the vibrant colors of your vegetables brighten and they become tender.

Remove the pot from heat and stir in 1 cup of whole milk or half-and-half for that creamy touch, mixing until well combined. If desired, add 1/4 cup of heavy cream for extra richness.

Finally, adjust the seasoning if necessary and ladle the hot soup into bowls. Garnish with chopped fresh parsley or chives and serve immediately.