Place the yeast in a large mixing bowl. Add the water and stir with a wooden spoon until the yeast is dissolved. (After a few minutes you should see bubbles appear on the surface of the yeast mixture; this will let you know that the yeast is working). Add 3 cups flour, 1 tablespoon of the olive oil, and the salt, and stir well to combine. Continue stirring until the dough pulls away from the sides of the bowl and comes together.
Place the dough on a lightly floured surface. Knead until it forms a smooth, elastic ball, 3 to 5 minutes.
Grease a large bowl with 1 tablespoon of the olive oil. Place the dough in the bowl, turning to coat with oil. Cover with a damp kitchen towel or plastic wrap. Place in a warm draft-free place and let rise until doubled in size, 1 to 2 hours.
Preheat the oven to 400?F.
When the dough has doubled, remove the towel and divide the dough in half. Keep a damp towel on the dough half that you are not working with and divide the portion of dough on your work surface into 12 pieces. Cover the dough pieces with a damp towel.
Line an 11 1/2 x 16-inch sheet pan with parchment paper. Working with 1 piece of dough at a time, roll the dough into a long thin rope by rocking your open palms over the dough in a back-and-forth motion. The dough rope should be 6 to 8 inches long. Use a lightly floured rolling pin to flatten and stretch the dough to about 12 inches long. Place 1 tablespoon of the chopped prosciutto along the center of the flattened dough, bring the edges of the dough up over the prosciutto, and pinch to seal. Using your fingertips and palms, gently roll the dough to make a log 13 to 14 inches long. Be careful-the seam where the dough was pinched may come undone. Simply pinch the dough again and gently roll it until it reaches the desired length.
Lay the breadstick on the parchment-lined sheet pan, and continue the process with the remaining 11 pieces of dough. Once you have all the breadsticks rolled and stuffed, use a pastry brush to brush 3 tablespoons olive oil over them. Season the breadsticks with cracked black pepper, and place the sheet pan in the oven. Continue making breadsticks with the remaining dough and prosciutto. Bake the breadsticks until light golden brown, about 20 minutes. Remove from the oven and serve hot or at room temperature.
Makes 24 breadsticks