Peel the cucumber into long ribbons. To do this, hold the cucumber and run a peeler the full length of the vegetable. Don't rotate the cucumber until you hit the seeds. This will produce wide ribbons. Discard the middle core of seeds.
In a large bowl, whisk together the rice vinegar, water, salt and sugar. Add the carrots and cucumber ribbons to the bowl. Set aside and let the vegetables pickle for 30 minutes.
In a large skillet over medium heat, cook the mushrooms with the sesame oil, sesame seeds and 2 tablespoons of soy sauce for 2 to 3 minutes. Transfer the mushrooms to a plate or bowl, and set aside. To the same pan, add the beef, ginger, hot chilli oil and remaining 1 tablespoon soy sauce. Stir to combine, and cook for 7 to 10 minutes or until the beef is cooked through and browned.
Spray a clean skillet with cooking spray, and set over medium heat. Gently crack the eggs into the skillet, cover with a lid, and cook for 3 minutes or until the egg whites are no longer clear but the yolk is still runny (sunny side up).
To assemble, place 1 cup of cooked rice in each of four bowls. Pour 1 tablespoon of the pickling liquid from the veggies over the rice. (For a spicier dish, mix 2 tablespoons sriracha with ½ cup pickling liquid and pour 1 tablespoon of that mixture over the rice.) Divide the beef, cucumber, carrots, and mushrooms evenly among the four bowls. Top each bowl with an egg, scallions, sriracha, and chilli toping, if used.