Preheat oven to 375 degrees. On lightly floured surface, roll each sheet of puff pastry into a 12-inch square. Lay 1 sheet puff pastry into a 10-inch pie plate; set aside.
In a 10-inch skillet, melt 1/4-cup butter until sizzling. Add potatoes, onion, 1/4 tsp. salt and 1/4 tsp. pepper. Cover and cook over medium-high heat, turning occasionally, until potatoes are lightly browned and crisply tender, about 12 to 15 minutes. Set aside.
In a new skillet, melt 1 tbsp. butter until sizzling. Meanwhile, in a small bowl, stir together all omelet ingredients except remaining 1 tbsp. butter. Pour half of omelet mixture (3/4 cup) into heated skillet. Cook over medium heat. As omelet mixture sets, lift slightly with spatula to allow uncooked portion to flow underneath. Continue cooking until set, about 2 to 3 minutes. Slide omelet onto cookie sheet. Repeat with remaining butter and omelet mixture.
Layer ingredients into pie plate on top of puff pastry in the following order: one of the omelets, 4 oz. ham, half of fried the potatoes, 1 cup shredded cheese, remaining potatoes, ham, cheese and 2nd omelet. Top with remaining sheet of puff pastry. Press together edges of both sheets of puff pastry to form a rim; trim off excess. Crimp or flute edges. In small bowl, stir together 1 egg and 1 tbsp. water; brush over puff pastry.
Bake 30 to 35 minutes or until golden brown. Let stand 5 minutes; cut into wedges.