Position rack in top third of oven; preheat broiler. Line a large rimmed baking sheet with foil.

Toss chicken, oil and ½ teaspoon salt on the prepared baking sheet until evenly coated. Arrange in a single layer, leaving space between each piece. Broil until lightly browned, about 7 minutes. Remove from oven; discard any excess liquid.

Meanwhile, whisk vinegar, mustard, brown sugar and the remaining ¼ teaspoon salt in a small saucepan; bring to a boil over high heat. Reduce heat to medium; simmer, stirring often, until thick and syrupy, about 5 minutes. Remove from heat. Whisk in butter until completely melted and the mixture resembles a caramel sauce. Transfer the mixture to a medium bowl, reserving 1 tablespoon for later use. Add the chicken to the balsamic mixture in the medium bowl; toss until evenly coated.

Arrange the chicken mixture in a single layer on the same baking sheet. Broil until glossy and deeply browned and an instant-read thermometer inserted into the thickest portion registers 165°F, about 3 minutes. Remove from oven; brush with the reserved 1 tablespoon balsamic mixture. Transfer to a platter and sprinkle with thyme.