Cut each rib crosswise with grain into 3 equal pieces. Slice each piece against grain ¼ inch thick. Toss beef with 2 tablespoons soy sauce, cover, and marinate for 10 minutes. Whisk remaining 1 tablespoon soy sauce, sesame oil, vinegar, and pepper flakes together in separate bowl; set aside.

Drain beef. Heat 1½ teaspoons coconut oil in 12-inch nonstick skillet over medium-high heat until just smoking. Add half of beef, breaking up any clumps, and cook, without stirring, for 2 minutes. Stir beef and continue to cook until browned, about 3 minutes; transfer to clean bowl. Repeat with 1½ teaspoons coconut oil and remaining beef; transfer to bowl.

Heat 1 tablespoon coconut oil in now-empty skillet over high heat until just smoking. Add mushrooms and cook until beginning to brown, about 2 minutes. Add bok choy stalks, cover, and cook for 3 minutes. Uncover and continue to cook, stirring frequently, until vegetables are spotty brown and crisp-tender, 3 to 4 minutes.

Push vegetables to sides of skillet. Add remaining 1 tablespoon coconut oil, ginger, and garlic to center and cook, mashing mixture into skillet, until fragrant, about 30 seconds. Stir ginger-garlic mixture into vegetables. Stir in bok choy greens and cook until wilted, about 2 minutes. Stir in beef and any accumulated juices. Off heat, add sauce and toss to coat. Sprinkle with scallions and serve.