Pat the chicken thighs dry with paper towels. Season both sides with kosher salt, black pepper, and dried oregano.
Heat the olive oil in a large 12-inch skillet over medium-high heat until shimmering.
Add the chicken thighs skin-side down and cook without moving for 6 to 7 minutes, until the skin is golden brown and crisp.
Flip the chicken thighs and cook for an additional 3 minutes. Remove the chicken from the skillet and set aside on a plate.
Reduce the heat to medium. Add the chopped onion to the skillet and sauté for 3 to 4 minutes until softened.
Add the minced garlic and cook for 30 seconds until fragrant.
Add the rinsed rice to the skillet and stir to coat the rice with the oil and onion mixture. Cook for 1 to 2 minutes, stirring frequently.
Pour in the chicken broth, lemon zest, lemon juice, ground cumin, and crushed red pepper flakes. Stir to combine.
Nestle the chicken thighs skin-side up into the rice mixture.
Bring the liquid to a simmer, then reduce the heat to low. Cover the skillet with a tight-fitting lid and cook for 25 to 30 minutes, until the rice is tender and the chicken is cooked through (internal temperature of 165°F).
Remove the skillet from heat and let it rest, covered, for 5 minutes.
Sprinkle chopped fresh parsley and crumbled feta cheese over the top before serving.