Place a medium saute pan over medium heat. Add the butter, garlic, thyme and rosemary and let cook until the butter melts, about two minutes. Meanwhile, season the cut sides of the courgettes and potatoes with salt and pepper. Carefully place the courgettes and potatoes cut side down in the melted butter. Let them cook until golden brown, about 12-15 minutes.
Preheat the grill. Line a baking tray with foil. Place the browned courgettes and potatoes on the baking tray cut side up. Sprinkle the tops with the Parmesan. Place under the grill until the cheese is golden brown, about four minutes. Transfer to a plate and serve.