Brush both sides of each pizza with the oil. Grill until golden brown on both sides, about 2 minutes per side. Remove from the grill.
Mix together the sugar, cinnamon and star anise in a small bowl. Brush the tops of each pizza with the butter and sprinkle with the sugar mixture. Place back on the grill, close the cover and cook an additional 1 minute. Cut into quarters and drizzle with the warm chocolate sauce.
In a large bowl stir together 1/3 cup water, yeast, and sugar and let stand until foamy, about 10 minutes. Stir in remaining 1/3 cup water, 2 tablespoons oil, 1 3/4 cups flour, cornmeal, and salt and blend until the mixture forms a dough. Knead dough on a floured surface, incorporating as much of remaining 1/4 cup flour, as necessary, to prevent dough from sticking, until smooth about 5 to 10 minutes. Form the dough into a ball.
Lightly oil the sides and bottom of a large bowl with oil, add the dough, turn to coat in the oil, cover and let rise in a warm place until doubled in size, about 1 hour. Gently punch the dough down and divide into 4 equal pieces. Roll each piece on a lightly floured surface into a 10-inch circle that is 1/8-inch thick. Brush off excess flour and transfer the dough to a baking sheet, cover each circle of dough with plastic wrap and continue stacking rolled out pieces on top of each other. Wrap well with plastic wrap and refrigerate until ready to grill.