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For the starter: Weigh your flour; or measure it by gently spooning it into a cup, then sweeping off any excess. In a small bowl, combine the flour, water, and yeast. Cover and let sit at room temperature for at least 8 hours, or overnight.
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To make the dough: In the bowl of your mixer or the bucket of your bread machine, combine the starter, water, semolina, yeast, and olive oil, stirring until the mixture is smooth.
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Add the flour, salt, cheese, and parsley or pizza seasoning, and mix until a soft dough forms. Knead the dough for 6 to 8 minutes until it's smooth and springy (or use the dough cycle on your bread machine).
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Let the machine finish its cycle, or place the dough in an oiled bowl, cover with greased plastic wrap, and let rise until doubled, 45 minutes.
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After the dough has risen, turn it out onto a lightly floured surface and divide it into 16 pieces. Roll each piece into a ball and let the dough rest for 10 minutes while you make sure the grill is hot and clean. You can refrigerate the dough, covered, for up to 3 hours at this point, if you need to adjust your timing or want to prep things ahead of time.
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To grill the bread: Flatten the balls of dough to 4" rounds about 1/4" thick. Spray the dough lightly with some olive oil, and place it on a medium-hot section of the grill. After 2 minutes, give the dough a quarter turn without flipping it over to make nice diamond-shaped grill marks. Cook another 2 minutes, flip the dough over, and cover with an aluminum pie plate or metal bowl if you have one. Cook another 2 to 4 minutes, then remove from the grill and serve. Repeat with the remaining pieces of dough.
