Pat chicken dry with paper towels; season with oregano, salt, and black pepper.
In a large, high-sided skillet over medium-high heat, heat oil. Add chicken skin side down and cook, turning occasionally, until golden brown, about 5 minutes per side. Transfer to a plate.
Reduce heat to medium and melt butter. Add garlic and cook, stirring, until fragrant, about 1 minute. Add broth, cream, lemon zest, lemon juice, and red pepper. Bring to a boil, stirring often, then add pasta and return chicken to skillet. Reduce heat to medium-low, cover, and cook until pasta is al dente and an instant-read thermometer inserted into thickest part of thigh registers 165°, 18 to 20 minutes.
Transfer chicken to a plate. Add feta to pasta mixture and stir until melted. Stir in parsley.
Return chicken to skillet. Top with more feta and parsley.
