Heat olive oil in a large deep skillet or Dutch oven over medium-high heat.
Season both sides of the chicken breasts with salt, black pepper, and dried oregano.
Add the chicken breasts to the skillet and sear for 4-5 minutes on each side until golden brown but not fully cooked through. Remove chicken from the skillet and set aside.
In the same skillet, add the chopped onion and cook for 3-4 minutes until softened.
Add the minced garlic and cook for 1 minute until fragrant.
Add the orzo pasta to the skillet and stir to coat it with the oil and onions for 1 minute.
Pour in the chicken broth and water, stirring to combine.
Return the chicken breasts to the skillet, nestling them into the liquid and orzo.
Bring the mixture to a boil, then reduce heat to low, cover, and simmer for 12-15 minutes, or until the orzo is tender and the chicken is cooked through (internal temperature 165°F).
Remove the chicken breasts from the skillet and set aside on a plate.
Stir in the heavy cream, lemon zest, lemon juice, and crumbled feta cheese into the orzo mixture until the sauce is creamy and the feta is slightly melted.
Return the chicken breasts to the skillet and spoon some sauce over them.
Sprinkle chopped fresh parsley over the dish before serving.