In a large pot over medium heat, melt the unsalted butter or olive oil. Add finely diced onion and minced garlic, cooking for about 3-4 minutes until the onion is translucent and fragrant.

Stir in the diced carrots and diced potatoes, sautéing for another 5 minutes until they begin to soften.

Sprinkle the all-purpose flour over the sautéed vegetables, stirring to coat them evenly. Cook for about 2 minutes.

Gradually whisk in the vegetable broth while ensuring no lumps remain. Bring to a gentle bubble and thicken.

Incorporate the dried thyme, dried rosemary, and season generously with salt and pepper. Allow to simmer for about 15 minutes.

Stir in the broccoli florets, frozen peas, and corn kernels. Continue to simmer for an additional 5-7 minutes.

Reduce the heat to low and gently stir in the whole milk or half-and-half.

Taste the soup and adjust seasoning as needed. Garnish with freshly chopped parsley before serving.