Heat the oven to 190 C / 375 F and grease a 23 x 33 cm / 9 x 13-inch baking dish. Cook the rice with 475 ml / 2 cups water and a pinch of salt in a covered 2-quart / 1.9 L saucepan for 15 minutes, then fluff with a fork. While it cooks, grate the cheddar, dice the onion, mince the garlic and cut the chicken into 2 cm cubes.
Pat the chicken dry and season with 3/4 tsp of the salt and the pepper. Heat 1 tbsp olive oil in a 30 cm / 12-inch skillet over medium-high heat until it shimmers. Sear half the chicken in a single layer for 4 to 5 minutes, turning once, until golden outside and not yet cooked through. Move it to the baking dish and repeat with the second half.
Microwave the broccoli with 2 tbsp water in a covered 2-quart / 1.9 L bowl for 2 minutes, until bright green and still firm. Drain and add it to the baking dish with the chicken and the cooked rice.
Return the skillet to medium heat and melt the butter. Cook the onion for 3 minutes until soft, then stir in the garlic and smoked paprika for 30 seconds. Sprinkle over the flour and stir for 1 minute. Pour in the milk slowly, whisking, and simmer for 3 minutes until the sauce coats the back of a spoon.
Take the skillet off the heat. Stir in the mustard, the remaining 1/4 tsp salt and 150 g of the cheddar until melted, then whisk in the cold yogurt until smooth. Keep the pan off the burner, because yogurt in a hot sauce curdles.
Pour the sauce over the chicken, broccoli and rice and fold gently until everything is coated. Smooth the top. Toss the panko with 1 tsp olive oil and the remaining 50 g cheddar, then scatter it over the dish.
Bake uncovered for 20 minutes, until the edges bubble, the panko is golden and a thermometer in the thickest chicken cube reads 165 F / 74 C.
Rest the casserole for 5 minutes so the sauce settles, then scoop with a wide spatula.