3 tablespoons extra-virgin olive oil
8 ounces fully cooked beer bratwurst, cut into 2-inch chunks
1 8 -ounce package sliced cremini mushrooms
2 stalks celery, cut into 1-inch pieces
1 carrot, cut into 1-inch pieces
1 bunch scallions, thinly sliced (white and green parts separated)
2 teaspoons chopped fresh thyme
1 1/2 cups long-grain white rice
2 1/2 cups low-sodium chicken broth
1/4 cup half-and-half
Kosher salt
1 cup grated Swiss cheese (about 4 ounces)