Whisk together stock, soy sauce, honey, sesame oil, and cornstarch in a bowl until cornstarch is dissolved. Season steak with salt and pepper. Heat oil in a large skillet over high heat. Add steak and cook, stirring occasionally, until almost cooked through, 2 to 3 minutes; transfer to a plate using tongs. Reduce heat to medium-high. Add broccoli and season with salt. Cook, stirring often, until broccoli is crisp-tender, 2 to 3 minutes (you can add a few tablespoons of water if pan starts to get too dark). Add ginger and garlic. Cook, stirring, until fragrant, 30 seconds to 1 minute.  Whisk sauce and add to pan with steak and any accumulated juices. Cook, stirring, until sauce is thickened and steak is warm, about 1 minute. Serve over rice topped with sesame seeds and scallions with lime wedges alongside.