How to Make It
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 and bagel chips, if desired, before serving.