Dissolve yeast and sugar in 3/4 cup warm water in a small bowl; let stand for 5 minutes.
Lightly spoon all-purpose flour into dry measuring cups; level with a knife. Combine 2 1/2 cups flour, 1 tablespoon stone-ground cornmeal, and 1 teaspoon salt in a large bowl; stir well with a whisk. Set aside.
Stir 1/2 cup warm sake into yeast mixture; immediately add to flour mixture, and stir until a soft dough forms. Turn dough out onto a floured surface. Knead until smooth and elastic (about 8 minutes); add enough of the remaining flour, 1 tablespoon at a time, to prevent dough from sticking to hands (dough will feel sticky).
Place dough in a large bowl coated with cooking spray, turning to coat top. Cover and let rise in a warm place (85°), free from drafts, 30 minutes or until doubled in size. (Gently press two fingers into dough. If indentation remains, the dough has risen enough.)
Preheat oven to 400°.
Turn dough out onto a baking sheet evenly sprinkled with remaining 1 teaspoon cornmeal. Gently press dough into a 12-inch circle. Lightly spray surface of dough with cooking spray; cover and let rise in a warm place (85°), free from drafts, 15 minutes.
Combine red onion, 1/2 teaspoon sake, olive oil, and dash of black pepper; toss well to coat. Combine 1 teaspoon water and egg white in a small bowl. Gently brush dough with egg white mixture; discard any remaining egg white mixture. Sprinkle fontina cheese evenly over dough, and top with the onion mixture. Sprinkle with thyme and 1 teaspoon kosher salt.
Bake at 400° for 35 minutes or until focaccia is deep golden brown and cheese is bubbly.