1Spray cookie sheets with the cooking spray. In a large bowl, place the yeast. In a 1-quart saucepan, heat the water over medium heat until warm and an instant-read thermometer reads 105°F to 115°F. Pour water over yeast; stir until yeast is dissolved. Stir in the sugar, 1 teaspoon salt, 1/4 cup oil and 1 cup of the flour. Beat with an electric mixer on medium speed until smooth. With a wooden spoon, stir in enough of the remaining flour, about 1/2 cup at a time, until dough is soft, leaves side of bowl and is easy to handle (dough maybe slightly sticky).
2Sprinkle flour lightly on a countertop or large cutting board. Place dough on floured surface. Knead by folding dough toward you, then with the heels of your hands, pushing dough away from you with a short rocking motion. Move dough a quarter turn and repeat. Continue kneading about 5 minutes, sprinkling surface with more flour if dough starts to stick, until dough is smooth and springy. If dough is sticky, lightly flour your hands. Shape dough into a uniform roll that's 10 inches long. Cut roll crosswise into 32 equal parts. Roll each part into a pencil-like rope, 8 inches long for thick breadsticks or 10 inches long for thin breadsticks. Place 1 inch apart on cookie sheets. Brush with 2 tablespoons oil. Lightly spray sheets of plastic wrap with cooking spray; cover dough loosely with the plastic wrap, sprayed side down. Let rise in a warm place 20 minutes.
3Place an egg separator over a small bowl. Crack open the egg, letting the yolk fall into the center of the separator and the egg white slip through the slots into the bowl. Cover and refrigerate egg yolk up to 4 days to use in another recipe.
4Move the oven rack to the middle position of the oven. Heat the oven to 350°F. In a small bowl, beat the egg white and 1 tablespoon water slightly with a fork or wire whisk. Brush mixture over breadsticks, using a pastry brush; sprinkle with coarse salt. Bake 1 cookie sheet at a time 20 to 25 minutes or until breadsticks are golden brown. Remove from cookie sheets to cooling rack. Serve warm or cooled.