Melt the butter in a large pot over medium heat. Add the onion and garlic, and sauté until they smell like they mean business—about 4 minutes.
Whisk in the flour. Keep stirring until it looks like a paste. No lumps, no panic.
Slowly pour in the broth while whisking. Then stir in the milk. Let it simmer for 5–6 minutes until it thickens just enough to coat the back of a spoon.
Toss in the broccoli. Let it cook uncovered for about 10–12 minutes, or until the broccoli is fork-tender.
Remove the pot from heat. Stir in the cheddar until melted and glorious. Add salt and pepper to taste.
Serve hot, with crusty bread if you're feeling fancy—or just a big spoon if you're not.