In a sauté pan over medium-high heat, warm the olive oil. Add the onions and sauté until golden brown, about 10 minutes. Add the water and continue to sauté until all the moisture evaporates, about another 5 minutes.

Add the Roquefort or blue cheese and continue cooking, stirring occasionally, until melted, about 5 minutes longer. Season only lightly with salt, if needed, and add pepper to taste. Stir in the walnuts and then spread the mixture out onto the chilled sheet pan. Place in the freezer until the onions cool down completely, about 10 minutes.

Preheat an oven to 450° F. Brush a baking sheet evenly with the melted butter.

Place the puff pastry on a cutting board. Using the rim of a small plate about 5 inches in diameter as a guide, cut the pastry into 8 rounds. Discard the pastry scraps.

Place 4 of the rounds on the prepared baking sheet. Brush their outer rims and tops with the beaten egg. Evenly distribute the cooled onion mixture onto the middle of each of the 4 rounds, leaving 1 inch uncovered around the edges. Place 1 slice prosciutto on top of each mound of the onion mixture. Cover each round with a second pastry round, making sure to pinch down firmly around the edges to seal in the filling. Brush the top of each pie with more of the beaten egg. Using a small, sharp knife, pierce the top of each pie once with a small slit.

Bake until golden brown, about 20 minutes. Remove from the oven and serve immediately.