Put figs, sugar, anise, 1 tbsp. thyme, and 1 cup wine in a medium saucepan. Cover and bring to a boil over high heat. Reduce heat and simmer until figs are just tender when pierced, 10 to 12 minutes. Let cool.
With a small, sharp knife, make 16 evenly spaced lengthwise cuts into roast, each cut about 1 in. long and 1 in. deep. Insert a garlic sliver, then a fig half into each cut, closing meat over figs; make cuts a little bigger if needed. Set aside remaining garlic and figs and their liquid.
Preheat oven to 325°. Using kitchen twine, tie pork crosswise at about 1 1/2-in. intervals and lengthwise twice to form a neat roast. In a small bowl, combine 1 tbsp. thyme, 1 1/2 tsp. salt, 1/2 tsp. pepper, and the oil. Rub all over roast.
Heat a 12-in. frying pan over medium-high heat. Brown pork all over, turning as needed, 8 to 10 minutes total; adjust heat if needed to keep meat from scorching. Transfer pork fat side up to a 9- by 13-in. baking pan.
Reduce heat to medium. Add reserved garlic to frying pan; cook, stirring often, until light golden, about 1 minute. Pour in remaining wine from bottle and bring to a boil, scraping up browned bits with a wooden spoon. Pour mixture over pork and cover tightly with foil.
Bake pork until almost tender when pierced, 2 1/2 hours. Stir reserved fig mixture into pan juices; bake, covered, until meat is tender, 15 to 20 minutes more.
Spoon pan juices over pork to moisten, then transfer meat to a cutting board and tent loosely with foil. Skim fat from pan juices. Pour juices with figs into a large frying pan and boil over high heat until reduced to 2 cups, about 5 minutes. Stir in 1/2 tsp. thyme. Taste and season with lemon juice and more salt and pepper if you like. Pour into a gravy boat. Remove twine from pork, then cut meat crosswise into thick slices. Garnish with thyme sprigs and serve with sauce.
Make ahead: Prepare through step 3 and chill airtight up to 1 day.
Note: Nutritional analysis is per serving.