Place 1/2 teaspoon of the saffron in a small bowl with the warm water and let it steep for 5 minutes.
In a large bowl, whisk the soaked saffron and water into the yogurt, then add the minced garlic, lemon juice, salt, and pepper until combined.
Add the chicken pieces to the marinade and toss thoroughly so each piece is well coated.
Cover and refrigerate the chicken for at least 30 minutes or up to overnight.
Preheat a grill to medium-high heat and oil the grates to prevent sticking.
Shake off excess marinade from the chicken, place pieces on the hot grill, and cook 15 minutes total, turning once or twice, until an instant-read thermometer registers 165°F (74°C).
While the chicken cooks, whisk the remaining 1/4 teaspoon saffron into the melted butter.
When the chicken is done, brush the saffron butter over the pieces and serve.