Preheat oven to 425°. Bring first 2 ingredients and 1 cup water to a boil in a large saucepan over high heat. Reduce heat to medium. Add flour all at once, and beat vigorously with a wooden spoon 1 to 2 minutes or until mixture is smooth and pulls away from sides of pan, forming a ball of dough. Remove from heat, and let stand 2 minutes. Stir in eggs, 1 at a time, beating vigorously with wooden spoon after each addition. (Mixture should be smooth and glossy.) Add Cheddar cheese, and stir until well blended.
Drop batter by level spoonfuls 1 inch apart onto a parchment paper-lined baking sheet, using a small cookie scoop (about 1 inch). Sprinkle with Parmesean cheese.
Bake at 425° for 10 minutes. Reduce oven temperature to 375°, and bake 17 to 20 more minutes or until golden brown.
Cut each gougère in half, and fill bottom half with desired amount of Deviled Ham. Top with remaining half.
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