Preheat oven to 400. Bring butter, salt and 1 C water to a boil in a large heavy pan over medium heat, stirring occasionally till butter melts. Add flour; stir until incorporated and continue to cook until mixture leaves the sides of the pan. Cool for 3 minutes.
Transfer dough to a large bowl; add eggs, one at a time, beating until incorporated. Stir in pepper, cheese and ham.
Spoon dough into a pastry bag -- I just dropped from a spoon. On a baking sheet lined with parchment paper, pipe out 1 1/2 inch wide rosettes about 1 inch apart. Make a deep indent in the center of each with a dampened thumb.
Bake until crisp and golden - 25-30 minutes. Transfer to a wire rack to cool. Press a cheese cube into each indentation when cool. Freeze until firm -- if desired.
To serve, preheat oven to 425 - place thumbprints on ungreased baking sheets until heated through and cheese cubes are melted - 10-14 minutes. Serve warm.