Preheat: Set your oven to 375°F (190°C).
Make Gravy Base: In a medium pot over medium heat, combine cream cheese, both cans of soup, and the Italian dressing mix. Stir until smooth.
Split Mixture: Pour most of the sauce into a large bowl (this is your filling base). Leave the remaining sauce in the pot on low heat to use as gravy later.
Combine Filling: Add the shredded chicken and green onions to the large bowl with the sauce. Mix well.
Assemble: Lay out crescent triangles on a cookie sheet. Place a spoonful of chicken mixture and a sprinkle of Monterey Jack cheese on the wide end of each triangle.
Roll: Fold the side corners in and roll toward the point to seal the bundle.
Bake: Bake for 9–12 minutes until the dough is golden brown.
Serve: Serve warm with the reserved gravy drizzled over the top.