Position a rack in the top third of the oven and preheat the broiler. Line a large rimmed baking sheet with foil. Toss the chicken pieces, canola oil, and 1/2 teaspoon salt on the prepared baking sheet until the chicken is evenly coated. Arrange the chicken in a single layer, making sure there is space between each piece. Broil for about 7 minutes until the chicken is lightly browned. Remove from the oven and discard any excess liquid.

While the chicken is broiling, whisk together the balsamic vinegar, Dijon mustard, light brown sugar, and the remaining 1/4 teaspoon salt in a small saucepan. Bring the mixture to a boil over high heat. Reduce the heat to medium and simmer, stirring often, until the sauce is thick and syrupy, about 5 minutes. Remove from heat and whisk in the butter until it's completely melted and the mixture resembles a caramel sauce. I like to let the sauce thicken just a tiny bit extra—this really helps it cling to the chicken later.

Transfer the balsamic sauce to a medium bowl, reserving 1 tablespoon of the sauce for later. Add the cooked chicken pieces from Step 1 to the bowl with the sauce and toss until the chicken is evenly coated.

Arrange the coated chicken mixture back in a single layer on the same baking sheet. Broil until the chicken is glossy, deeply browned, and an instant-read thermometer hits 165°F, about 3 minutes. Remove from the oven, brush the chicken with the reserved 1 tablespoon balsamic sauce, then transfer to a platter. Sprinkle with fresh thyme leaves to finish.