In a large skillet, heat olive oil and butter over medium heat. When heated, put chicken skin-side down in the pan, apply salt and pepper, and cook for 5 minutes.
Flip chicken, apply salt and pepper to other side, and cook an additional 5 minutes. Dice onion and prepare mushrooms in this time.
Remove chicken from the skillet and add the onion and mushrooms with an additional pinch of salt. Cook 2-3 minutes until tender, then add thyme leaves, mustard, and heavy cream. Whisk to combine well, return chicken to the pan, cover, and cook 15-20 minutes until temperature reads ~175°, stirring occasionally to prevent sauce burning.
Bring a large skillet of salted water boiling. When the chicken has about 10 minutes left, put the pasta in the boiling water and cook until al dente. Drain and return pasta to skillet.
Transfer chicken from sauce to top of pasta, then in the sauce skillet off the heat, add the capers, juice of the lemon, and parmesan cheese. Whisk to combine then pour over chicken and pasta. Toss to coat and serve.
