1 tablespoon olive oil
1 medium onion, chopped (could use 2-3 green onions as a substitute)
1/8 teaspoon crushed red pepper (optional)
2 cups low-sodium vegetable broth
1 bunch broccoli, florets roughly chopped and stems peeled and sliced (about 7 cups)
1 large russet potato (about 8 ounces), peeled and cut into 1/2-inch pieces
salt and black pepper
½ cup sharp white Cheddar, grated
Heat the oil in a large saucepan over medium heat. Add the onion and red pepper (if using) and cook, stirring occasionally, until the onion is soft, 4 to 6 minutes.
Add the broth, broccoli, potato, 2 cups water, ½ teaspoon salt, and ¼ teaspoon black pepper to the saucepan and bring to a boil. Reduce heat and simmer, covered, until the vegetables are very tender, 18 to 20 minutes.
In a blender, working in batches, puree the soup until smooth, adjusting the consistency with water as necessary.
(Alternatively, use a handheld immersion blender in the saucepan.) Top with the Cheddar, before serving.