Make the dough: Preheat oven to 350 degrees. Coat 4 baking sheets generously with olive oil and set aside. Place the flour, cheese, sage, baking powder, pepper, salt, and paprika in the bowl of a food processor fitted with a metal blade and pulse to combine. Add 1/4 c olive oil and pulse until the dough resembles coarse meal. Gradually add ice water and continue to pulse until dough forms a ball and rides the blade.
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Shape the breadsticks: Transfer the dough to a floured surface. Roll the dough into a 16-by-8-inch rectangle using a well-floured rolling pin. Cut the dough into sixteen 1-by-26-inch strips with a sharp knife and cut each strip in half. With the palms of your hands. Roll each strip into about a 16-inch-long breadstick. Transfer breadsticks to the prepared baking sheets, brush generously with remaining oil, and sprinkle with coarse salt.
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