Crank the oven to 425°F (218°C) and position racks in the top third and bottom third positions of the oven. Line 2 baking sheets with parchment paper or nonstick baking sheets.
Bring the water, butter, and salt to a boil in a medium saucepan over medium-high heat.
Dump the flour all at once into the boiling butter mixture and, using a wooden spoon and good old elbow grease, stir until the flour clumps into a ball and pulls away from the sides of the pan. Continue stirring over medium-high heat until the dough leaves an even, dry film on the bottom of the pan, 2 to 3 minutes. (Don’t you dare skimp on this step.)
Drop the ball of dough into the bowl of a stand mixer fitted with the paddle attachment and mix on low speed until the dough is just barely warm to the touch, about 5 minutes.
Plop in an egg, increase the speed to medium, and beat until incorporated. The dough may look hopelessly soupy and broken at first, but fear not and continue mixing until the blob turns smooth and slick. Repeat with the remaining eggs, adding them 1 at a time.
Using a spoon, stir in the Prosciutto di Parma, Grana Padano, rosemary, thyme, and pepper.
Drop rounded tablespoons of the dough, or use a 1-tablespoon cookie scoop to make plump mounds, on the prepared baking sheets, spacing them about 2 inches apart. Have a glass of cold water handy and dip the spoon in it every so often to help make the dough easier to scoop.
Bake the puffs until golden brown, 20 to 25 minutes, rotating and switching the positions of the sheets once halfway through baking.
Transfer the sheets to a wire rack to cool slightly. Then whip off your apron, pop as many puffs as can fit into your mouth, place the rest of the puffs on a platter, and offer them to your guests post haste. (You can bake these up to 2 hours in advance, then reheat them in a 350°F [176°C] oven for 5 to 7 minutes. Or freeze a batch of baked, cooled puffs for up to a month and, just before guests arrive, pop the frozen puffs into a 350°F [176°C] oven for 10 to 12 minutes.)