Preheat the oven to 450 degrees. Toss the carrots, onion and garlic in vegetable oil and roast until very pierced easily with a fork, about 20 minutes (they should be softer than if you were just going to eat them roasted).
Add everything from the tray, including any oil/juices left on the pan, into a stock pot or Dutch oven. Add the vegetable broth, and bring to a simmer over medium heat.
Being extremely careful of the hot soup, turn off the burner and blend using a blender or a stick blender until smooth. Add the coconut milk and use the blender to mix that in also. You may need to work in batches.
Add the soup back into the stock pot/Dutch oven and add the salt and grated ginger. Taste, and judge thickness.
If the soup needs to be thicker (I like a good, thick soup), ladle about a cup out to another bowl and whisk in the cornstarch until dissolved. Add the mixture back into the pot and bring to a boil for a full minute. You can repeat, if you'd like to thicken the soup further.