Heat oven to 375°F. Set out a 9-by-13-inch baking dish. Use fully cooked chicken that was originally cooked to 165°F and refrigerated promptly if made ahead. Steam broccoli for about 3 minutes until barely tender, then drain very well.

Melt butter in a large saucepan over medium heat. Cook onion for 5 minutes until softened. Add garlic and cook 30 seconds. Stir in flour and cook, stirring, for 1 minute.

Gradually whisk in milk and broth. Bring to a gentle simmer, stirring, and cook for 3–5 minutes until the sauce coats a spoon. Remove from heat. Stir in half the cheddar, salt and pepper.

Fold chicken and drained broccoli into the sauce. Transfer to the baking dish and scatter remaining cheddar over the top. Bake uncovered for 25–30 minutes, or longer as needed, until the center reaches 165°F and the edges bubble.

Let stand 5 minutes, then divide into six portions. Refrigerate leftovers in shallow containers within 2 hours, or 1 hour above 90°F, at 40°F or below. Use within the original cooked chicken’s 3–4-day refrigerated window and reheat throughout to 165°F.