Cook the udon noodles according to package instructions, then drain and set aside.
In a large skillet or pot, heat the vegetable oil over medium heat. Add the sliced onion and sauté for about 2 minutes until translucent.
Add the sliced carrot and mushrooms to the skillet, cooking for an additional 3-4 minutes until the vegetables are tender.
Pour in the vegetable or chicken broth and bring to a simmer. Stir in the curry powder, soy sauce, and sugar, mixing well to combine.
If using, add the cooked protein to the skillet and heat through for about 2 minutes.
Add the cooked udon noodles to the skillet, tossing to coat them in the curry sauce. Cook for another 1-2 minutes until everything is heated through.
Serve hot, garnished with chopped green onions.