Preheat a grill to medium. Whisk the yellow mustard, vinegar, brown sugar, mustard powder, hot sauce, Worcestershire sauce, 1/2 teaspoon salt, and pepper to taste in a bowl. Whisk in the butter.

Season the chicken with salt and pepper, then toss with about one-third of the mustard sauce in a large bowl until coated. Let sit at room temperature, 10 minutes.

Brush the grill grates generously with vegetable oil. Put about 1/4 cup of the remaining mustard sauce in a small bowl for basting and reserve the rest for topping.

Grill the chicken, covered, basting occasionally with the sauce, until well marked and a thermometer inserted into the thickest part registers 170 degrees F, 10 to 12 minutes per side.