Preheat oven to 350°. In a Dutch oven, heat butter over medium-high heat. Add onion and garlic; cook and stir 2-3 minutes or until tender. Stir in flour, white pepper and salt until blended; gradually whisk in milk. Bring to a boil, stirring constantly; cook and stir 2 minutes or until thickened.
Stir in 1 cup cheese. Reduce heat; cook 1-2 minutes or until cheese is melted (sauce will be thick). Remove from heat.
Add potatoes, ham and broccoli to sauce; stir gently to coat. Transfer to eight greased 8-oz. ramekins.
Bake, covered, 40 minutes. Sprinkle with remaining cheese. Bake, uncovered, 20-25 minutes longer or until potatoes are tender and cheese is melted.
