Sauté aromatics
In a large pot, melt butter over medium heat. Add onion and cook until soft. Stir in garlic and cook for 30 seconds.
Cook potatoes
Add diced potatoes and broth. Bring to a boil, then reduce heat and simmer for 15 minutes or until potatoes are tender.
Add broccoli
Stir in broccoli and cook for 5 minutes, just until tender and bright green.
Blend
Use an immersion blender to partially blend the soup, leaving some texture.
Add dairy & cheese
Lower heat. Stir in milk, then gradually add cheese, stirring until melted and smooth.
Season & serve
Season with salt and pepper. Serve warm with your favorite toppings.