For the bread: Preheat the oven, pizza oven or grill to 550 degrees F. Add 1/2 cup of the bread flour, the lukewarm water and the yeast to a stand mixer fitted with the paddle attachment. Mix well and let sit until bubbly, about 30 minutes.
Whisk together the remaining 3 1/2 cups bread flour and the salt in another bowl so it will be evenly distributed.
Once the yeast mixture is nice and bubbly and looks like foamy beer, add 3/4 cup cold water and the olive oil. Fit the mixer with a dough hook and begin mixing, adding the flour mixture in increments. Mix the dough until the dough starts creeping up the dough hook and comes away from the sides of the bowl, about 5 minutes.
Place the dough in a large bowl lightly greased with olive oil. Turn to coat all sides of the dough with oil. Cover the bowl loosely with a clean tea towel. Let rise until doubled in size, about 2 hours. Punch the dough down, and let rise for another hour.
Divide the dough into 4 equal disks. Lightly flour a work surface with semolina flour. Stretch each disk out to a 10-inch round using your fingers or heels of your hands (and a rolling pin, if you prefer).
Flatten the disks of dough into rounds about 1/2-inch thick and press with fingers to make holes. Drizzle with extra virgin olive oil and sprinkle with sea salt.
If using a pizza oven, bake until the bread has browned and is starting to char slightly on the edge, about 10 minutes.
If using a grill, lay the dough oiled-side up and grill until char marks appear on the bottom. Flip and cook for another 2 minutes on the second side.
If using a regular oven, turn a baking sheet upside-down on the center rack of the oven. Slide the schiacciata onto the baking sheet. Bake until browned on the edges, 10 to 15 minutes. (Ovens vary, so times will be different. Use the visual clues to help you know when it's done.)
Set the breads aside to cool.
For assembly: Slice the schiacciata in half horizontally to make sandwich halves. Cut into quarters and fill with assorted fillings as desired. Serve immediately.