In a large pot, heat oil and sauté the chopped onion along with the carrots but keep stirring as you don't want these burnt. Add the garlic and sauté for 30 more seconds.
Add potatoes, chicken broth, and season with salt and pepper. Let it boil, then cover with a lid and lower the heat. Cook for about 15 minutes. Add broccoli and cook for 5 more minutes minutes.
Add milk, flour, and thyme to the soup, let it boil until veggies are soft and tender. Serve warm and top with cheddar cheese if desired.