Preheat oven to 375°F (190°C) and grease an 8x10-inch baking dish.
In a skillet, heat olive oil and sauté onion for 3–4 minutes; add garlic and cook 30 seconds.
Add flour, stir, then whisk in broth until smooth; simmer 2–3 minutes to thicken.
Remove from heat, stir in Greek yogurt and most of the cheeses until creamy.
Combine chicken and broccoli in a large bowl; pour sauce over and mix gently.
Transfer to baking dish, top with remaining cheese.
Mix breadcrumbs with oil and Parmesan, sprinkle on top.
Bake 25–30 minutes until bubbly and golden. Let rest 5 minutes before serving.