Trim excess fat from the chicken thighs and cut each thigh in half; set aside.
Peel the onions and slice into slivers; finely chop the parsley and set aside.
Heat the olive oil and butter in a large heavy frying pan over medium-high heat until hot.
Brown the chicken quickly on both sides in the hot pan, about 2–3 minutes per side; do not cook through. Transfer chicken to a plate.
Add the sliced onions to the pan (add a bit more oil or butter if needed) and cook over medium heat until edges are golden, about 8–10 minutes; transfer browned onions to a plate.
Warm the chicken broth until very warm, then stir in the saffron to bloom.
Return the chicken to the pan, arrange the browned onions over the chicken, and pour the saffron-infused broth over everything.
Cover the pan and simmer very gently over low heat for 15–20 minutes without turning the chicken, until cooked through.
Uncover, add the chopped parsley and lemon juice, and add a tablespoon or two of water only if the pan looks dry; simmer uncovered about 10 minutes more to meld flavors.
Serve the chicken hot.