Heat oil in a large pot over medium heat. Add onion, curry paste, ginger, and garlic. Stir continuously until onion begins to soften, about 2 minutes. Pour in vegetable broth and coconut milk. Add sugar. Bring to a boil. Stir in tofu, broccoli, cauliflower, and mushrooms.
Reduce heat and cover pot. Simmer soup, stirring occasionally, until vegetables are tender, 5 to 8 minutes. Season with salt and pepper.