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 and set vegetables aside.
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 sweet red chili sauce on the side for mixing into bibimbap.