We can do this two ways. You can dollop each individual crescent and roll them up the traditional way. Or you can leave two crescents together, pinch the seams at the perforation, making a rectangle and pile double the amount in. Then just gently pull the corners up and pinch it perfect.
Preheat the oven to 350 degrees F. Line a rimmed baking sheet with parchment paper.
Heat the oil in a medium nonstick skillet over medium-high heat. Sprinkle the chicken with some salt and pepper and sear on one side for 5 minutes. Flip and sear for 5 minutes more. Transfer the chicken to a plate to let cool for 5 minutes, then give it a good chop.
Meanwhile, add the broccoli to a food processor and pulse until finely minced. While the chicken is resting, add the broccoli to the skillet with a pinch of salt and saute until the broccoli slightly wilts and pops in color, 3 to 5 minutes, adding a little oil if needed.
Combine the chicken, broccoli, mozzarella, mayonnaise, garlic powder, Italian seasoning and a pinch of salt in a large bowl until fully incorporated.
Roll the crescents out two ways: You can unfold each individual triangle and dollop about 1 tablespoon of the chicken mixture at the thicker end, rolling it up to make sure the tip is on top. Or you can unroll two crescents together to make a rectangle, pinch them together at the perforation, dollop about 1/2 cup of the chicken mixture in the center and gently pull the corners together on top, pinching to seal.
Place the crescent rolls on the prepared baking sheet and brush with the beaten egg. Sprinkle each with the Parmesan and bake until golden brown on top, 14 to 20 minutes depending on their size. Serve with any dipping sauces you prefer.
For the grated mozzarella, make sure to use a block, not fresh. It melts much better. I used about half an 8-ounce block of cheese, but I honestly could have gone for more.