Heat sesame oil in a large skillet over medium heat; cook and stir carrot and zucchini in the hot oil until vegetables begin to soften, about 5 minutes. Stir in bean sprouts, bamboo shoots, and mushrooms. Cook and stir until carrots are tender, about 5 more minutes. Season to taste with salt; transfer vegetable mixture to a plate.

Stir cooked rice, green onions, soy sauce, and black pepper in the same skillet until the rice is hot.

In a separate skillet over medium heat, melt butter and gently fry eggs, turning once, until the yolks are still slightly runny but the egg whites are firm, about 3 minutes per egg.

To serve, divide hot cooked rice mixture between 3 serving bowls and top each bowl with 1/3 of the vegetable mixture and a fried egg. Serve with sweet red chili sauce.