Place 1 ¼ cups warm water in large bowl. Sprinkle yeast and sugar over; stir to blend. Let stand until yeast dissolves, about 10 minutes. Mix in cornmeal, oil and salt.
Add 3 cups flour to yeast mixture; stir until dough forms. Turn dough out onto floured work surface. Add remaining ¼ cup flour; knead until dough is smooth and elastic, about 2 minutes.
Lightly oil clean large bowl. Add dough; turn to coat. Cover with plastic wrap, then kitchen towel. Let dough rise in warm draft-free area until doubled in volume, about 45 minutes.
Punch dough down. Turn out onto floured surface; knead briefly until smooth. Return to oiled bowl; turn to coat. Cover with plastic, then kitchen towel. Let dough rise 40 minutes. Punch dough down. Turn out onto floured surface. Divide into 4 pieces. Cover with plastic; chill at least 1 hour or overnight.
Preheat oven to 450*F (230*C). Oil 11 x 15-inch baking sheet. Roll out 1 dough piece to 1/8-inch-thick rectangle. Transfer to prepared baking sheet. Using fingers, gently stretch dough to edges of baking sheet. Bake until golden, puffed and crisp, about 8 minutes. Remove from oven. Top with ½ cup mozzarella, ¼ cup Gorgonzola, ¼ cup Parmesan and 1 teaspoon garlic. Return cracker to oven; bake until cheese melts and bubbles, about 4 minutes. Remove from oven. Sprinkle with 1 tablespoon each of parsley and green onions. Remove cracker from baking sheet. Cool baking sheet. Repeat with remaining dough, cheeses, garlic, parsley and onions. Break warm crackers into pieces.
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