Add the rice to the skillet and cook over medium heat, stirring occasionally, until fragrant and lightly toasted (about 1 minute).
Simmer the Rice and Chicken:
Stir in the chicken broth, milk, and the remaining 1 teaspoon kosher salt. Bring to a boil, stirring occasionally to prevent sticking.
Lower the heat to medium-low, cover, and cook for 5 minutes.
Add the chicken pieces, stirring to distribute evenly. Cover and continue to cook, stirring occasionally, until the rice is tender and the chicken is cooked through (10 to 12 minutes).
Combine and Broil:
Stir the caramelized onions and half of the shredded Gruyère (about ¾ cup) into the skillet. Taste and adjust the seasoning with more salt and pepper if needed.
Sprinkle the remaining Gruyère evenly over the top.
Broil to Perfection:
Preheat the broiler to HIGH with a rack positioned 4 inches from the heat source.
Broil the dish until the cheese is melted and golden-brown in spots (3 to 4 minutes).
Serve and Garnish:
Garnish with additional thyme leaves and a sprinkle of black pepper.
Serve hot and enjoy!