For the dough: In a small bowl, sprinkle the yeast over 1/4 cup of the lukewarm water and let stand 2 minutes to soften. Whisk with a fork to dissolve the yeast, and let stand 5 minutes to activate.
In the bowl of a stand mixer, combine the yeast-water mix, bread flour, semolina flour, butter, olive oil, salt and remaining 1 1/4 cups water. Mix with a dough hook on medium speed for about 10 minutes; the dough will clear the sides of the bowl. Alternatively, mix the dough by hand with a wooden spoon, then knead by hand for 10 minutes. Shape the dough into a ball and place in an oiled bowl. Turn to coat the dough with the oil, cover the bowl with plastic wrap, and let the dough rise until doubled, about 1 hour.
For shaping: Combine the bread flour, semolina flour and salt in a small bowl and stir to blend. In another small bowl, combine the olive oil and water.
Lightly oil four heavy-rimmed baking sheets (or bake in batches).
Sprinkle the work surface thickly with the flour mixture. Turn the dough out onto the work surface and flatten it with a rolling pin into an 18- x 6-inch rectangle. If the dough feels too soft and slack to shape the breadsticks, transfer the dough to a floured sheet pan and place in the freezer for about 10 minutes to firm it. Return it to the floured work surface before continuing.
Brush the surface of the rectangle with the oil-water mixture, then sprinkle generously with some of the flour mixture. With a chef's knife, cut the dough into 6-inch by 1/4-inch strips, cutting just a few at a time. With your hands, pick each strip up by the ends. The dough is so supple that it will elongate as you lift it. Allow each strip to become only as long as the baking sheet, then arrange them side by side on the baking sheet, close but not touching. Let rise about 30 minutes.
Preheat oven to 350°. Bake the breadsticks until caramel-brown all the way through (test by breaking one open), about 30 minutes. Cool on a rack.
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