Stir the dark soy sauce, soy sauce, sugar, and chile-garlic sauce together in a small saucepan and place over medium-how heat; simmer and stir until the sugar dissolves into the sauce, about 5 minutes. Remove from heat and set aside.
Heat the olive oil in a skillet over medium heat. Cook and stir the garlic and chicken in the hot oil until the chicken is no longer pink in the center, 7 to 10 minutes. Stir the broccoli into the chicken mixture; cook and stir until the broccoli is thoroughly heated. Add the noodles and stir until all the ingredients are evenly mixed. Pour the sauce over the mixture and stir until everything is evenly coated; continue cooking until the sauce begins to thicken.
Push the chicken mixture to the side of the skillet with a spatula. Add the egg to the skillet in the vacated space. Scramble the egg, cooking it through. Once egg is cooked, mix ingredients back in and heat thoroughly. Garnish with the sesame seeds and red pepper flakes to serve.