Place the flour, yeast, sugar, 2 teaspoons salt, half the rosemary and garlic in the food processor bowl. Process in a few bursts to mix. With the processor running, slowly add 1 cup of the water, then the 1/2 cup olive oil and finally the remaining 3/4 cup water. Process until the dough coalesces into a ball and starts riding around on the blade. Turn out on a floured board and knead a few strokes. Dough should be relatively stiff but still pliable. Place in an oiled bowl, cover with plastic wrap and allow to rise in a warm place 'til doubled, about 1-1/2 hour.
Place the baking stone in the oven and preheat to 450º. Punch the dough down, divide into 2 balls and flatten each ball to a disk about 8 - 10 " in diameter. Place on a corn-meal covered board or peel, cover with a clean towel and allow to rise in a warm place for about 1/2 hour.
When the dough has finished the second rising, use your forefinger to poke deep dimples all over the loaf, about 1-1/2" apart. Drizzle olive oil over the loaves, scatter on the remaining rosemary & garlic then slide them onto the stone. Bake 10 minutes, then lower the oven temperature to 375º and bake 20 - 25 minutes more, until the bread is a nice dark golden brown. Two minutes before the bread is done, brush with the egg wash and sprinkle generously with the coarse salt.
Allow to cool on a rack for 5 - 10 minutes before serving.