In a small bowl add the soy sauce, brown sugar, garlic, ginger, green onion, sesame oil, chili paste, and ground black pepper, mix to combine, and set aside.
Cook the udon noodles according to package instructions. Rinse, drain, and set aside in a bowl with a cover to keep the heat and moisture in.
Heat 2 tbsp of olive oil in the cast iron pan over high heat. Add the Chinese eggplant and stir to coat in the oil. Cook for 5-6 minutes until tender and slightly browned. Remove from the pan into a separate bowl and cover to keep the heat and moisture in.
Add another tbsp of olive oil to the hot skillet along with the broccoli and carrots. Cook for 2-3 minutes until soft but not too tender. Remove from the pan and add to the bowl with the eggplant, covered.
Turn the heat down to med-high and add 1 more tbsp of olive oil (be careful the pan is very hot at this point). Add the slices of beef and cook them while stirring constantly. They will cook very fast, only about 1-2 minutes depending on how thick the slices are.
Add the vegetables, noodles, and sauce to the pan with the beef and stir together until everything is covered in the sauce. Serve immediately in bowls and sprinkle with sesame seeds if desired. Enjoy!