Dissolve honey in warm water in a large bowl; sprinkle yeast over top. Let stand until yeast softens and begins to form a creamy foam, about 5 minutes.

Stir in onion, 1 tablespoon olive oil, 1 tablespoon salt, then add flour until dough comes together. Turn dough out on a well-floured surface; knead until smooth and elastic, about 5 minutes.

Place dough in a lightly oiled large bowl and turn to coat. Cover with a damp cloth and let rise in a warm place until doubled in volume, about 20 minutes.

Preheat the oven to 415 degrees F (215 degrees C). Oil a baking sheet.

Place dough onto the prepared baking sheet; flatten to evenly cover the whole sheet. Press indentations into dough spaced about 1 inch apart using fingertips. Drizzle with remaining 3 tablespoons olive oil, then sprinkle Parmesan cheese, rosemary, and remaining 1 tablespoon kosher salt over top. Let rise for 10 minutes.