Combine the beef, hoisin sauce, soy sauce, scallions, half each of the ginger and garlic, and 1/2 teaspoon each salt and pepper in a bowl. Heat 1 tablespoon vegetable oil in a large skillet over high heat. Add the beef mixture and cook, breaking up the meat, until browned, 5 minutes.
Add the mushrooms to the skillet; cook, stirring, until wilted, 1 to 2 minutes. Transfer the mixture to a bowl. Wipe out the skillet, then heat the remaining 1 tablespoon oil over medium-high heat. Add the remaining ginger and garlic and cook until golden brown and crisp, 2 to 3 minutes; set aside.
Bring 4 cups water to a boil in a large pot over high heat. Add the bok choy and ramen noodles along with the reserved ramen flavor packet. Cover and cook, stirring halfway through, until the boy choy is wilted and the noodles are tender, about 4 minutes. Add the beef-mushroom mixture to the pot and toss to combine. Top each serving with the ginger-garlic mixture and the sliced chile.