Fill a pot with water and add two large pinches of salt. Bring to a boil over high heat. Add the pasta and cook until tender, about 8-10 minutes. Drain and rinse briefly with cool water.
While pasta cooks, combine olive oil, balsamic vinegar, mayonnaise, grated Parmesan, and Dijon mustard in a jar. Shake until well combined.
Roughly chop the spinach, slice the grape tomatoes in half, and dice the mozzarella into 1/2-inch cubes.
In a large bowl, combine the drained pasta, chopped spinach, halved tomatoes, and diced mozzarella.
Toss gently to combine. Add half the dressing and toss again. Adjust dressing to taste. Serve immediately or store for later.