Put ½ cup pomegranate juice and 1/8 cup balsamic vinegar into a small saucepan. Bring to a boil over medium-high heat, then reduce to low and simmer, stirring occasionally, until the liquid has reduced and thickened enough to coat the back of a metal spoon, about 10–15 minutes. Turn off the heat and set the sauce aside to cool (it will thicken further as it cools).

In a small bowl, stir together 4 tbsp pesto sauce and 1 tbsp olive oil until smooth.

Pat the 8 small lamb rib chops dry with paper towels. If using, season both sides lightly with salt.

Spread the pesto–olive oil mixture evenly over both sides of each lamb chop.

Heat a cast-iron grill pan over high heat until very hot. Place the lamb chops in a single layer without crowding. Sear the first side 2–3 minutes (use 2 minutes for a less-cooked result, 3 minutes for more color), then flip and sear the second side 3 minutes for medium/medium‑well or 4 minutes for well‑done.

Transfer the lamb chops to a serving platter and let rest a few minutes. Spoon the cooled pomegranate–balsamic sauce over the chops and serve.