Season and sear the chicken: Heat the oven to 350°F and grease a 9x13 baking dish. Toss the chicken with 1/2 teaspoon of the salt, the black pepper, and the paprika. Heat the olive oil in a large Dutch oven over medium-high heat and cook the chicken 6-7 minutes, turning the pieces, until browned outside and 165°F on a thermometer. Move it to a plate.

Build the base: Melt the butter in the same pot over medium heat. Add the onion and cook 3 minutes until soft, then stir in the garlic and thyme and cook 30 seconds more.

Simmer the rice: Add the rice, chicken broth, and the remaining 1/2 teaspoon salt. Stir once, bring it to a boil, then reduce the heat to low and cover. Simmer 7 minutes, then scatter the broccoli over the top without stirring, cover again, and simmer 8 minutes more.

Rest off the heat: Take the pot off the burner and leave it covered for 10 minutes. Don't lift the lid. The rice finishes absorbing the last of the broth during those minutes.

Stir it together: Uncover and fold in the milk, parmesan, 1 cup of the cheddar, and the cooked chicken along with any juices on the plate. It should look loose and saucy, not dry.

Fill the pan: Spread everything into the greased 9x13 dish and scatter the remaining 1 cup of cheddar over the top.

Bake: Cover with foil and bake 20 minutes, then uncover and bake 10 minutes more, until the cheese is melted with browned spots and the edges bubble. Let it stand 5 minutes before serving.