Line 2 large rimmed baking sheets with parchment or wax paper. In a food processor, work the soppressata with the Pecorino, mustard and water to a fine paste. Transfer the paste to a bowl.
On a lightly floured work surface, roll out the puff pastry 1/8 inch thick. Spread the soppressata paste evenly over the bottom half of the dough, leaving a 1/4-inch border on the sides. Fold the top half of the dough over to cover the paste and press the edges to seal. Using a rolling pin, lightly press the top and bottom together. Slide the puff pastry onto a lined baking sheet and chill until firm.
Roll out the pastry a scant 1/8 inch thick and cut into 3 long 6-inch wide strips. Working with 1 strip at a time and keeping the others refrigerated, trim the pastry to a neat rectangle. Brush with the egg wash and sprinkle with one-third of the sesame seeds. Cut the dough crosswise into 3/4-inch-wide sticks and arrange them 1/2 inch apart on the prepared baking sheets. Refrigerate until chilled.
Preheat the oven to 375°. Bake the soppressata sticks for about 16 minutes, or until golden and crisp, shifting the pans halfway through baking. Serve warm or at room temperature.
Make Ahead: The baked sticks can be kept in an airtight container for up to 4 days or frozen for up to 2 weeks. Recrisp the sticks in a 350° oven.
Wine Recommendation: Bobby Stuckey: 1997 was a banner year in Piedmont, Italy, and that region's Francesco Boschis Dolcetto di Dogliani Sorì San Martino bursts with fruit--a great balance to spicy cheese sticks.Madeline Triffon: Spice, salt and dairy fat--all the flavors that tame astringent tannins--call for a big red like the 1995 Nottola Vino Nobile di Montepulciano Vigna del Fattore from Tuscany.