Heat the oil and butter in a deep-sided, thick-bottomed pan over medium to medium-high heat. When the butter is melted, add the chicken skin side down. Cook 12-15 minutes until brown. Flip. Cook another 12-15 minutes until cooked through.
Remove the chicken and cover to keep warm.
Discard ~half the oil. Add the shallots to the pan and brown. Add the wine and cook until reduced by half. ~10 minutes. Add the heavy cream. Cook until it turns light brown, ~5 minutes. Return the chicken to the pan to coat in the sauce. Garnish with parsley and serve!