Bring a large pot of water to a boil. Add the broccoli and cook until crisp-tender, 2 minutes. Remove with a slotted spoon to a plate. Return the water to a boil and cook the noodles according to package directions.
Meanwhile, cut the steak in half lengthwise. Cut crosswise into 1/4 inch slices. Mix the hoisin sauce, lime juice, chili-garlic paste, and cornstarch in a small bowl until smooth.
Spray a large non-stick skillet (or wok) with nonstick spray and set over a medium-high heat. Add one-third of the beef and cook, stirring constantly, just until browned, 1-2 minutes. Transfer to a plate and repeat with the remaining beef.
Add the bell pepper and onion to the skillet. Reduce the heat and cook, stirring constantly, until crisp-tender, 5 minutes. Add the ginger and cook, stirring constantly, until fragrant, 1 minute. Add the hoisin sauce mixture and cook, stirring frequently, until the sauce simmers, 1 minute. Add the beef and broccoli, cook, stirring frequently until heated through, 2 minutes.
Drain the noodles. Rinse under cold water until cool and drain. Transfer 1 1/2 cups of the beef mixture and 2 cups of the noodles into separate containers and let cool. Cover and refrigerate up to 3 days for later use. Add the remaining 2 cups of noodles to the skillet and cook until heated through, 1 minute.