Trim any excess fat from the chicken thighs and cut each thigh in half.
Peel the onions and slice them into slivers; finely chop the parsley and set aside.
Heat the olive oil and butter in a heavy frying pan over medium-high heat until shimmering.
Brown the chicken pieces briefly on both sides, about 2–3 minutes per side, without cooking through; remove to a plate.
Add the onions to the pan (add a little more oil or butter if needed) and cook over medium heat until edges turn golden, about 8–10 minutes; remove onions to a plate.
Warm the chicken broth until very warm and stir in the saffron to infuse.
Return the browned chicken to the pan, arrange the cooked onions over the chicken, and pour the saffron-infused broth over everything.
Cover the pan and cook over a very low simmer for 15–20 minutes without turning the chicken, until cooked through.
Uncover, add the chopped parsley and lemon juice, and if needed add a splash of water; simmer uncovered for about 10 minutes more to meld flavors.
Serve the chicken hot.