Melt the unsalted butter or olive oil in a large pot over medium heat. Add the diced onion and minced garlic, stirring occasionally for 3–4 minutes until fragrant.

Stir in the diced carrots and potatoes, sautéing for an additional 5 minutes.

Sprinkle the all-purpose flour over the vegetables, stirring well for 1–2 minutes until lightly golden.

Gradually whisk in the vegetable broth, breaking up any lumps, and whisk for 2–3 minutes until smooth.

Add thyme, rosemary, salt, and pepper. Bring to a simmer, cover, and cook for 15 minutes.

Stir in the broccoli florets, frozen peas, and corn. Simmer uncovered for 5–7 minutes.

Mix in the milk or half-and-half, and heavy cream if using. Stir gently for 2 minutes over low heat.

Ladle soup into bowls and garnish with chopped parsley before serving.