Warm the oil in a large pot over medium heat. Cook the onion until soft and translucent, about 5 minutes, then add the 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.
Add the broccoli and simmer just until bright green and fork-tender, about 5 minutes. Do not overcook it or the color goes dull.
Blend about half the soup until smooth, either with an immersion blender or in batches, then return it to the pot. The potato does the thickening here.
Lower the heat, stir in the milk and dry mustard, then add the cheddar a handful at a time, stirring until each addition melts before adding more. Season and serve.