1. In a medium bowl, stir together yeast, warm water, and honey. Let rest until yeast blooms and bubbles form on top, about 10 minutes.
2. Stir in flour, 1/4 cup olive oil and 1 teaspoon kosher salt. Turn the dough onto a well-floured surface and knead until dough is smooth, 5 to 10 minutes. Place dough in a lightly oiled bowl, cover with a kitchen towel or plastic wrap, and let rest in a warm place until dough doubles in size, about 1 hour.
3. Preheat oven to 450°F. Remove dough from bowl and press it into a lightly oiled 9- by 13-inch baking sheet until it touches the edges. Using your finger, poke holes all over the dough. Drizzle the dough with 2 tablespoons olive oil. Let rest until the dough becomes puffy, about 20 minutes.
4. Heat 1 tablespoon olive oil in a large skillet set over medium heat. Add onion slices, cover and cook until onion is golden brown, stirring occasionally, about 20 minutes.
5. Top the dough with tomato slices, caramelized onions, rosemary, Parmesan cheese, and salt. Drizzle with remaining 2 tablespoons olive oil.
6. Bake until the focaccia is golden brown, about 20 minutes. Remove from oven and allow to cool on a rack. Cut into pieces and serve.