Prep the Lamb: Trim the lamb of all connective tissue and excess fat. Reserve the trimmings for the pan sauce the next day.
Make the Marinade: Mix together all of the marinade ingredients. Pour the marinade into a resealable bag or container.
Marinate the Lamb: Add the lamb and give it a toss to coat it in the marinade. Put it in the refrigerator overnight. Take the lamb out of the refrigerator 1/2 hour - 1 hour before you are going to grill.
Preheat the grill.
Start the Pan Sauce: While the grill is heating, begin making the pan sauce. In a medium skillet on medium high heat, saute the lamb trimmings until they are fully cooked and crispy. Scoop them out of the skillet, leaving the fat and the bits of meat stuck to the pan behind.
Add the salt, shallots and garlic and saute for 30 seconds. Add the port and bring to a boil. Let the port reduce for 5 minutes, then add the chicken broth and balsamic vinegar. Bring to a boil again and reduce for 5 minutes. Take the sauce off the heat while you grill the lamb.
Grill the Lamb: Take the lamb out of the marinade and wipe it off. You don't want all that olive oil on the grill.
Grill the lamb on high for 3-4 minutes per side. Medium rare lamb is 125F, and medium is 135F. Use your meat thermometer to check for doneness. Set the lamb aside to rest for a few minutes while you finish up the sauce.
Finish the Sauce: On medium heat, bring the sauce back up to a low simmer. Add the figs. Warm them through, 1-2 minutes.
Take the sauce off the heat and add the thyme. Wait about 30 seconds, then start stirring in the butter. Once the butter has melted and the sauce is glossy, serve the fig port sauce with the lamb.