In a medium-sized mixing bowl, combine all of the ingredients except the paprika to make a cohesive dough, sprinkling in a tablespoon or so of water if the dough doesn't seem to want to come together.
As soon as the dough starts to come together, turn off the mixer and gather it into a rough ball.
Transfer it to a lightly floured work surface, and roll it into a 16" log about 1 1/2" in diameter.
Wrap the log in waxed paper or plastic wrap, and chill it in the freezer for 30 minutes.
Using a serrated knife, slice the log crosswise into 1/8" rounds. Place them on an ungreased or parchment-lined baking sheet, leaving only about 1/2" between them; they won't spread much as they bake. Sprinkle them with a bit of paprika, if desired.
Bake the cheese pennies in a preheated 400°F oven for 11 to 13 minutes, or until they're just beginning to brown. Remove them from the oven, and allow them to cool on the pan for several minutes before transferring them to racks to cool completely.
Store, well-wrapped, for up a week.