Bring a pot of salted water to a boil over medium-high heat. Add asparagus and cook until soft, about 5 minutes. Drain and rinse under cold water. Drain well.
Whisk mustard, balsamic vinegar, and white wine vinegar together in a small bowl. Add olive oil and whisk until well combined. Season with salt and pepper. Add a little bit of lukewarm water if dressing is too thick.
Combine asparagus, tomatoes, and mozzarella in a bowl and drizzle with dressing. Mix well and sprinkle with basil.