In a food processor blend the flour, the salt, the Cheddar and the butter until the mixture resembles meal. With the motor running add 5 tablespoons of the water and blend the mixture, adding more water if necessary, until it just forms a dough. Do not overblend the dough. Transfer the dough to a sheet of wax paper and halve it.
In the food processor grind fine the pepperoni with the cayenne, add half the dough reserving the other half, chilled, and blend the mixture until the pepperoni is just distributed throughout the dough. On the sheet of wax paper halve the pepperoni dough and divide 1 of the halves into thirds, reserving the other half, wrapped and chilled. Roll each third into an 8-inch rope. Repeat the rolling procedure with half of the reserved plain cheese dough.
On a new sheet of wax paper arrange the 6 ropes side by side, alternating the pepperoni with the plain, press them tightly together, and top them with another sheet or wax paper. With a rolling pin flatten the ropes between the sheets of wax paper and roll them into 1/8-inch-thick rectangles (about 15 by 7 inches). Transfer the dough and the wax paper to a baking sheet and chill it for at least 1 hour or overnight. Repeat the entire procedure with the reserved plain cheese and pepperoni doughs. The sheets of dough may be made 1 week in advance and kept covered tightly and chilled.
Cut the sheets of dough crosswise into 1/2-inch-wide strips, discarding the wax paper. Bake the strips in batches on greased baking sheets in the middle of a preheated 400 degree oven for 10 minutes, or until they are golden, transferring them as they are baked to racks, and let them cool. Serve the straws freshly baked.
Makes about 50 straws.