Preheat oven to 375°F (190°C). In a bowl, season shredded chicken with half the minced garlic, ½ teaspoon salt, and ¼ teaspoon pepper. Set aside.
In a measuring cup, whisk cornstarch with ½ cup of the chicken broth until smooth. Add remaining broth and stir until well combined.
Grease a 9x13 inch (23x33 cm) baking dish with olive oil. Layer uncooked pasta evenly across the bottom of the dish.
Arrange seasoned chicken over the pasta layer, then distribute broccoli florets evenly on top.
Pour the cornstarch-broth mixture and skimmed milk over the layers. Add remaining garlic, salt, and pepper. Gently press down with a spatula to ensure pasta is submerged.
Cover tightly with aluminum foil and bake for 25 minutes.
Remove foil, sprinkle with ¼ cup Parmesan cheese, and bake uncovered for an additional 5 minutes until cheese melts.
Remove from oven and let rest for 5 minutes. Stir in Greek yogurt and remaining Parmesan cheese until well combined.