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Pile all the flour on a clean work surface and make a well in the center. In the center of the well, place the butter, 2 eggs, egg yolk, and pinch of salt. Knead all ingredients together to form a smooth, soft ball of dough. Divide the dough into 2 pieces, 1 slightly larger than the other. Roll the larger piece out until it is large enough to line the bottom of an 11-inch tart pan. Prick it with a fork all over, then roll out the second piece of dough to approximately the same size as the first; it will be smaller but should fit over the tart pan like a lid.
Distribute the ham evenly over the bottom layer of dough. Top with the mozzarella, then add the beaten eggs, reserving 1 to 2 tablespoons, for brushing the top of the pastry. Top with Parmigiano-Reggiano and fit the "lid" over the top, sealing the edges all the way around with finger pressure.
Brush the top with the reserved egg and bake until golden brown, about 30 minutes. Let rest 10 minutes before slicing into wedges and serving like pizza.