Warm oil in a large pot over medium heat. Cook onion until soft and translucent, about 5 minutes; add garlic and cook 30 seconds more.
Add the potato and broth. Bring to a simmer and cook until the potato is completely tender, 12 to 15 minutes.
Simmer broccoli just until bright green and fork-tender, about 5 minutes. Avoid overcooking to preserve color.
Blend about half the soup smooth with an immersion blender or in batches; return to pot. Potato thickens it.
Lower heat; stir in milk and dry mustard. Add cheddar by handfuls, stirring until each melts before adding more. Season and serve.