Put the watercress and spinach in a large bowl, cover with a damp paper towel, and refrigerate.
In a small bowl, whisk the minced scallions, vinegar, hoisin sauce, soy sauce, and ginger. Whisk in the sesame oil and vegetable oil. Season to taste with salt and pepper.
Heat a large, heavy pan (preferably cast iron) over medium-high heat for at least 1 min. Use paper towels to pat the steaks dry. Season them generously with salt and pepper. Rub the steaks with oil, put them in the pan and cook them until well browned, about 4 min. Flip the steaks and continue to cook, about another 3 min. for medium rare. Transfer the steaks to a cutting board, let them rest for 5 min., and then slice them thinly.
Add the water chestnuts to the bowl of greens. Whisk the vinaigrette and toss the greens and water chestnuts with just enough of it to coat (there should be a few tablespoons vinaigrette left over). Taste and add salt and pepper as needed. Portion the greens and water chestnuts among four plates. Arrange the beef over the greens, drizzle with any remaining dressing, and serve.