Preheat the oven to 400 degrees F. Heat the olive oil in a large ovenproof skillet over medium-high heat. Add the bratwurst, mushrooms, celery, carrot, scallion whites and thyme. Cook, stirring occasionally, until the mushrooms soften, about 4 minutes.
Add the rice and cook, stirring to coat, 1 minute. Add the chicken broth, half-and-half and 1 teaspoon salt. Bring to a simmer, then cover the skillet and transfer to the oven. Bake until the rice is tender, about 18 minutes.
Switch the oven to broil. Uncover the rice and sprinkle with the cheese. Broil until the cheese is browned and bubbling, about 2 minutes. Top with the scallion greens. Photograph by Antonis Achilleos