Place an oven rack in the upper-middle position and preheat to 400°F. In a small bowl, whisk together the condensed soup, Greek yogurt, and Dijon; set aside.
Pat the chicken dry and season with 1/4 teaspoon salt and a pinch of black pepper. Heat a large oven-safe skillet (11–12 inches) over medium-high heat and melt 2 tablespoons butter.
Add onion with a small pinch of salt and cook until translucent, 2–3 minutes. Stir in garlic for 30 seconds. Push the aromatics to the edges, add the chicken to the center in a single layer, and sear until mostly opaque with light browning, 3–4 minutes. Transfer chicken and any juices to a plate (it will finish cooking later).
Reduce heat to medium. Add the remaining 1 tablespoon butter to the skillet, then stir in the rinsed rice and smoked paprika. Cook, stirring, for 60–90 seconds to lightly toast the grains.
Pour in the chicken broth, scrape up any browned bits, and season the liquid with the remaining salt and pepper. Bring to a simmer, then cover, reduce heat to low, and cook for 12 minutes.
Scatter the broccoli evenly over the rice (do not stir). Cover and cook until the broccoli is crisp-tender and the rice is just cooked, 6–8 minutes. If the pan looks dry, splash in 2–3 tablespoons broth or water.
Take the skillet off the heat. Fluff the rice, then fold in the reserved chicken and any juices. Pour in the soup-yogurt mixture and half of the cheddar. Stir until the casserole is creamy and evenly combined; taste and adjust seasoning as needed.
Smooth the top, sprinkle with the remaining cheddar, and bake uncovered for 8 minutes. Switch to broil for 1–2 minutes until the cheese is spotty and golden. Let rest 5 minutes before serving.