Prep vegetables: Wash and chop potatoes, broccoli, onion, and garlic into bite-sized pieces.
Sauté aromatics: Melt butter in a large pot, add onion, and cook until softened.
Make roux: Add garlic and flour, stirring constantly for one minute until fragrant.
Add liquids: Slowly whisk in broth, then add potatoes and bring to boil.
Cook potatoes: Simmer 15 minutes until potatoes are fork-tender throughout.
Add broccoli: Stir in broccoli florets and cook 5 more minutes until bright green.
Finish soup: Add milk and bring to a gentle simmer, then stir in cheese gradually.
Season and serve: Add salt and pepper to taste, let rest 5 minutes before serving. Top with chedder extra cheese, if you desired, and serve warm.