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Preheat the oven to 425°F. Line two baking sheets with parchment paper.
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In a medium-sized saucepan set over medium heat, bring the water, butter, salt, and pepper to a boil, boiling and stirring until the butter is completely melted.
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Add the flour all at once and stir over the heat until the mixture forms a smooth ball; this should only take 20 seconds or so.
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Remove the pan from the heat, and transfer the dough to the bowl of your electric mixer. (If you have a hand mixer, you may leave the dough in the saucepan.)
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Beat in the Parmesan cheese, then add the eggs to the dough one at a time, beating well after each addition. The dough will become very smooth and shiny. Since eggs, even within a size range, vary quite a bit, check the dough after you've added 3 eggs. If it's smooth, shiny, and has a pudding-like consistency (e.g., if you were to drop a tablespoonful onto a baking sheet, it would be fairly smooth, but would hold its shape nicely and not spread), then don't add the remaining egg.
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Using a tablespoon cookie scoop or a spoon, drop large chestnut-sized (1 1/4") balls of dough the baking sheets, leaving about an inch between them. The dough will be very sticky; it helps to dip your scoop or spoon in water every so often.
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Wet your finger and smooth the tops of the puffs.
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Bake the puffs for 35 to 45 minutes, until they're puffed and golden brown.
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Remove them from the oven, pierce the top of each carefully with the tip of a sharp knife, and serve hot, warm, or at room temperature. They'll be crispy-crunchy, with soft insides, hot from the oven; they'll gradually soften all over as they cool.
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Freeze for up to a month. To reheat, cover with aluminum foil and bake in a preheated 350°F oven for 8 to 10 minutes.
