Cook rice - For white rice, rinse until clear. Fill a saucepan or Dutch oven (or rice cooker) with rice, water (use 2:1 ratio for water to rice), and some salt. Cover with lid and bring to a boil. Once water boils, stir, lower heat to a low-medium, and cover. Check white rice after ~20 min; brown at ~45 min. Rice is done when it's absorbed all the water. If it's not done, leave the cover on until all water has been absorbed. Take cover off to release steam and fluff with a wooden spoon after 5 minutes. (Can be done up to 5 days ahead)
Marinade - Grate ginger and chop garlic. Combine with soy sauce, brown sugar, black pepper, sesame oil and red pepper flakes.
Beef - Slice against the grain into 2” (5 cm) long strips. Season with salt and pepper and tenderize with a fork. Pour half the marinade over beef (reserve the other half to cook with) and marinate for at least 30 minutes and up to a day. (Can be done up to 1 day ahead)
Whisk water and cornstarch together. Add to reserved half of marinade.
Heat a wok over medium-high heat. Add first part of cooking oil. When it begins to shimmer, add beef. Saute until browned on the outside, ~2 minutes. Remove from wok and set aside (it will finish cooking in the sauce).
Return wok to heat and add second part of cooking oil. Add peppers and broccoli. Cook until tender, 3 to 4 minutes. Add reserved half of marinade and bring to a simmer. Add beef and cook until beef is cooked through and sauce coats vegetables, ~2 minutes. Remove from heat and top with a squeeze of lime juice.
If rice was prepared in advance, reheat in microwave.