1. For the starter: In a small bowl, combine the flour, water, and yeast. Cover and let sit at room temperature for at least 8 hours, or overnight.
2. 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.
3. Add the flour, salt, cheese, and parsley, 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).
4. 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. It should look like this.
5. 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.
6. 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.