This recipe makes two 12-inch pizzas. One pound of store-bought whole-wheat pizza dough made with olive oil can be substituted for homemade dough if desired. If King Arthur flour is not available in your area, substitute 1 cup whole-wheat flour mixed with 1 3/4 cups unbleached all-purpose flour.
In a food processor fitted with the plastic blade blend the flour, yeast, and salt. In a 2-cup measuring cup mix the water, olive oil, and honey. With the food processor running add the water-oil mixture and blend until the flour forms a ball of dough. Process for 1 minute to knead the dough. The dough will be a bit sticky, but if too wet, add up to 1/2 cup more flour. Spray a bowl with nonstick cooking spray. Put the dough into the prepared bowl, cover with plastic wrap and let rise in a warm place until the dough is doubled in size about 1 hour.
In a large skillet over medium heat, heat 2 tablespoons olive oil. Add the onion, salt, and pepper. Cook, stirring occasionally until golden brown, about 15 minutes. Remove to a bowl. In the same skillet, heat 1 tablespoon olive oil. Add the eggplant and salt and cook stirring for 2 minutes. Cover and cook stirring occasionally for 5 minutes to soften the eggplant. Uncover and cook for 2 to 3 more minutes to remove any excess moisture. Remove to a bowl. In the same skillet heat 1 tablespoon olive oil. Add arugula and cook, tossing the arugula until the it is wilted. Remove from the skillet and set aside.
To assemble the pizzas, preheat the oven, and a baking stone if desired, to 425°F. Punch down the dough and divide into two pieces. Set on a lightly floured surface and cover with a towel to rest for 5 minutes. For each pizza: sprinkle a baking sheet with 1 teaspoon cornmeal. Roll one piece of dough into a 12-inch circle and place on the prepared baking sheet. Brush dough with 1 tablespoon olive oil. Distribute 1/2 cup caramelized onions, 1/2 cup cooked eggplant, and 1/4 cup arugula on the dough. Sprinkle each with 2 tablespoons shredded cheese. Repeat with other piece of dough. Bake the pizzas for about 15 minutes until the crust is lightly browned. Cut each pizza into 8 slices.