Preheat oven to 375°F. Lightly grease a 9x13-inch baking dish.
Steam broccoli until crisp-tender, 2–3 minutes. Drain well and pat dry.
Warm olive oil in a skillet over medium heat. Soften onion 4 minutes, then stir in garlic.
Whisk broth, milk, and cornstarch in a saucepan. Simmer, whisking, until slightly thick, 3–4 minutes.
Remove from heat. Whisk in yogurt, Dijon, lemon zest, paprika, salt, pepper, and red pepper flakes.
Combine chicken, rice, broccoli, and sautéed onions in a large bowl.
Pour sauce over mixture and toss to coat. Fold in half the cheddar.
Spread into the dish. Top with remaining cheddar, Parmesan, and panko.
Bake until bubbly and lightly golden on top, 22–25 minutes.
Rest 5 minutes. Scatter parsley and serve warm.