Finely chop steak into bitesize pieces. Combine beef and arrowroot starch in a medium bowl, tossing to coat. Heat peanut oil in a medium skillet over medium-high. Add beef; cook, stirring occasionally, until well browned and crispy, about 6 minutes. Add garlic, ginger, pepper, and 1/4 teaspoon of the salt; cook until ginger is softened, 1 to 2 minutes. Remove from heat; stir in 1 tablespoon of the vinegar.
Whisk together sesame oil, honey, chili-garlic sauce, and remaining 2 tablespoons vinegar and 1/4 teaspoon salt in a small bowl.
Separate leaves from bok choy, and place tender green tops in a medium bowl. Thinly slice white bottoms to equal 1 cup, and add to bowl. Using a peeler, shave carrot into long strips to equal 1 1/2 cups. Diagonally slice celery stalk into thin pieces to equal 1/2 cup, and chop celery leaves. Add carrots, celery, and celery leaves to bowl. Drizzle with sesame oil mixture, tossing to coat. Divide mixture evenly among 4 bowls; sprinkle with beef mixture and peanuts.