Pat the chicken dry with paper towels and season with oregano, salt, and pepper.
In a large skillet over medium-high heat, warm the olive oil. Add the chicken, skin side down, and cook until golden brown on both sides, about 5 minutes per side. Transfer the chicken to a plate.
Reduce the heat to medium and melt the butter. Add the garlic and cook until fragrant, about 1 minute. Stir in the chicken broth, cream, lemon zest, lemon juice, and red pepper flakes. Bring to a boil, then add the pasta and return the chicken to the skillet.
Reduce the heat to medium-low, cover, and simmer until the pasta is al dente and the chicken reaches an internal temperature of 165°F, about 18 to 20 minutes.
Transfer the chicken to a plate. Stir the feta cheese into the pasta mixture until melted, then add the parsley.
Return the chicken to the skillet and top with more feta and parsley before serving. Enjoy your delicious one-pan lemon butter feta chicken pasta!