In a medium-large pot with a tight fitting lid (dutch oven is best), warm the oil over medium heat. Add the turkey and ½ teaspoon salt and cook, breaking up into small pieces, until cooked through, about 3 minutes.
Stir in the garlic, oregano, and red pepper flakes (if using) and cook for about 30 seconds, stirring often.
Stir in the rice to coat in the oil, then add the water, mustard, soy sauce, and remaining ½ teaspoon salt. Stir well to combine.
Cover with a lid, bring to a boil over high heat, then simmer over low or medium-low heat for 12 minutes.
Remove the lid, scatter the broccoli evenly on top and simmer, covered, for 5 more minutes or until the broccoli is cooked but still has a little crunch.
Add 1 cup cheese and use a fork to fluff it into the rice. Top with the remaining 1 cup cheese and cover to let the steam melt it, about 1-2 more minutes.
Serve topped with black pepper.