In a large saucepan, heat the olive oil over medium heat. Add the sliced onions and sugar, stirring occasionally until the onions are caramelized and golden brown, about 15-20 minutes.

Stir in the thyme and cook for an additional minute before removing the onions from the pan and setting aside.

In the same saucepan, melt the butter over medium heat. Add the rice and salt, stirring to coat the rice in the butter.

Pour in the vegetable broth and bring to a boil. Once boiling, reduce the heat to low, cover, and simmer for 15-20 minutes, or until the rice is tender and the liquid is absorbed.

Once the rice is cooked, fluff it with a fork and gently fold in the caramelized onions.

Serve warm, topped with grated Parmesan cheese and chopped parsley if desired.