In a large shallow bowl, stir together sugar and salt. Press the cut side of each fig into the sugar mixture to coat.
Heat a large skillet over high heat and add sugar-covered figs, cut side down, and cook until sugar and juices caramelize, about 3-4 minutes. As figs caramelize, turn them over.
Remove skillet from the heat and add balsamic vinegar. Return to heat and cook to reduce the vinegar a bit, roughly 30 seconds, making sure to scrape up any caramelized bits on the bottom of the pan.
Remove from heat again and add butter and lemon juice, gently swirling to combine.
Serve figs immediately or wait until they cool to room temperature. Add to your cheese plate or top ice cream with the caramelized figs.