Preheat the oven to 425°F (220°C). Scrub the Yukon Gold potatoes and cut them into evenly sized 3/4-inch (2 cm) cubes. Place them in a large mixing bowl. Add the olive oil, kosher salt, black pepper, garlic powder, and smoked paprika. Toss thoroughly until the potato pieces are evenly coated with oil and seasoning. The surfaces should look lightly glossy, with no large patches of dry seasoning.

Spread the seasoned potatoes across a large rimmed baking sheet in a single layer. Leave a little space between the pieces so hot air can circulate and moisture can escape. Roast at 425°F (220°C) for 25 minutes, turning the potatoes about halfway through. The potatoes should be lightly golden around the edges and nearly tender when pierced with a fork. They will finish cooking during the final casserole bake. If the potatoes still feel very firm, continue roasting briefly until nearly tender.

While the potatoes roast, cut the cooked chicken into bite-sized pieces if it has not already been diced or shredded. Cut the fresh broccoli into small, relatively uniform florets, trimming away tough stem ends. Place the broccoli in a microwave-safe bowl with a microwave-safe cover and microwave for 2 minutes, just until bright green and beginning to soften. Drain any accumulated moisture thoroughly. The broccoli should remain slightly firm because it will continue cooking in the casserole. Keep the prepared chicken and broccoli ready for assembly.

Place a medium saucepan over medium heat and add the unsalted butter. Once the butter has melted, sprinkle in the all-purpose flour. Whisk continuously for 1 minute, until the mixture forms a smooth, lightly bubbling paste. The mixture should remain pale rather than turning dark brown. This flour-and-butter base will thicken the milk into a creamy sauce.

Gradually pour the whole milk into the saucepan, whisking constantly to prevent lumps. Continue cooking over medium heat for 3 to 4 minutes, stirring frequently. The sauce should become smooth and thick enough to lightly coat the back of a spoon. If small lumps appear, whisk vigorously until the mixture becomes uniform. Reduce the heat to low once the sauce has thickened.

Add the sour cream, onion powder, and kosher salt to the warm sauce. Stir until evenly combined. Gradually add the 1 cup (113 g) of shredded sharp cheddar reserved for the sauce, stirring until melted before adding more. The finished sauce should look creamy, smooth, and thick but still pourable. Avoid allowing the sauce to boil after adding the cheese and sour cream, as excessive heat can cause the dairy to separate or become grainy.

Reduce the oven temperature to 375°F (190°C). Transfer the roasted potatoes to the large mixing bowl. Add the cooked chicken and prepared broccoli. Pour the cheddar cheese sauce over the ingredients and gently fold everything together using a large spoon or spatula. Mix until the potatoes, chicken, and broccoli are evenly coated. The mixture should look creamy, with distinct pieces of potato and broccoli still visible. Avoid vigorous stirring, which can break apart the potatoes.

Transfer the mixture to a 9 x 13-inch (23 x 33 cm) baking dish. Spread it into an even layer, making sure the chicken and broccoli are distributed throughout the casserole. Sprinkle the remaining 1/2 cup (57 g) of shredded sharp cheddar cheese evenly over the surface. The cheese should form a light, even topping rather than a thick, compact layer.

Place the baking dish on the center oven rack. Bake uncovered at 375°F (190°C) for 20 minutes. The cheese should be completely melted, with small bubbles appearing around the edges and a few lightly golden spots on top. Check the center with an instant-read thermometer. Because the casserole contains previously cooked chicken, the center should reach 165°F (74°C) before serving. If the topping begins browning too quickly before the center is hot, loosely cover the dish with foil and continue baking until the proper temperature is reached.

Remove the casserole from the oven and let it rest for 5 minutes. This short resting period allows the sauce to settle slightly, making the casserole easier to portion. Sprinkle with the optional chopped fresh chives just before serving. Spoon into individual portions while warm. The finished casserole should have tender potatoes, moist chicken, broccoli that still holds its shape, and a creamy cheese sauce throughout.