Position oven rack about 8 inches from broiler and preheat broiler to high. In a large bowl, whisk eggs and milk until well combined. Whisk in salt and pepper.
Melt butter in a 10-inch nonstick ovenproof skillet over high heat. Add ham; sauté until just beginning to turn golden, 3 to 5 minutes.
Add corn and thyme and sauté until corn is warmed through, about 2 minutes longer. Reduce heat to medium-low, pour in egg mixture, top with scallions and cheese and cook, stirring, for 2 minutes. Stop stirring and cook undisturbed until bottom is just firm, about 5 minutes.
Transfer skillet to oven and broil until top is set and edges are barely beginning to brown, 1 to 2 minutes, watching carefully to prevent overcooking. Remove from oven, cover and let stand for 5 minutes. Serve warm or at room temperature.