Combine 1 cup flour, undissolved yeast and salt in a large mixer bowl. Heat water and oil until very warm (120o to 130oF); stir into dry ingredients. Stir in basil, oregano and enough remaining flour to make soft dough. Knead dough on a lightly floured surface until dough is smooth and elastic, about 8 to 10 minutes. Cover; let rest on floured surface 10 minutes.
While dough rests, heat 1 tablespoon olive oil in a medium skillet and sauté onion rings until softened. Set aside for topping.
Divide dough into 8 equal portions; form each into smooth ball. Roll each ball to 4-inch round. Place on greased baking sheets.
Use a wooden spoon to make indentations in dough at 1-1/2-inch intervals. Brush focaccia evenly with 2 tablespoons oil. Top each with cheese, artichokes, olives, mushrooms and sautéed onions, dividing evenly. Cover; let rise in warm, draft-free place until doubled in size, about 1 hour.
Bake at 400oF for 18 to 22 minutes, or until done. Serve immediately.
Note: Focaccia can be reheated in 350oF oven for 5 to 10 minutes. If needed, cover loosely with foil to prevent additional browning.