Place onion, red pepper, sausage, and beer in a large saucepan. Bring to a boil and cook 15 minutes.
Preheat the oven to 400 degrees F. Heat the olive oil in a large ovenproof skillet over medium-high heat. Add the celery, carrot, green onion whites and cook 3 minutes. , Add the bratwurst, mushrooms, and thyme. Add 1 tbsp oil if needed. Cook, stirring occasionally, until the mushrooms soften, about 4 minutes. Season with salt.
Add the rice and cook, stirring to coat, 1 minute. Add the chicken broth and half-and-half. Bring to a simmer, then cover the skillet and transfer to the oven. Bake until the rice is tender, about 18 minutes.
Uncover. Add cheese and broil until it melts. Remove from oven and top with green onion greens.