Preheat the oven to 200 degrees . Line a baking sheet with parchment. Place the dough onto a floured work surface; press to flatten. In a medium skillet, melt 2 tbsp. butter over medium-high heat. Add the onion. Season and cook, stirring, until soft, 5 minutes; cool.
Stretch the dough into a 10-inch square. Sprinkle with the cooked onion, parmesan and parsley. Roll up jellyroll- style, starting at the short end; cut crosswise into 8 slices. Place on the baking sheet, spacing evenly. Turn off the oven; let the dough rise in the oven for 15 minutes. Remove the baking sheet from the oven; place in a warm spot to rise for another 15 minutes. Increase the oven temperature to 400 degrees .
Bake the rolls until they are golden and sound hollow when tapped, about 20 minutes. Melt the remaining 2 tbsp. butter and brush on top of the rolls. Sprinkle lightly with salt. Let cool at least 15 minutes before serving.