Put figs, sugar, anise, 1 tablespoon 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 and then chop.
Chop garlic and add ½ of the garlic and ½ of the figs in a bowl and stir to combine.
Keep remaining garlic and figs separate.
Using a sharp knife cut along the side of the shoulder bone, cut ¾ through the meat and open out the meat like and book.
Take 4-5 strings of bakers twine so that it is spaced about 1 ½” intervals lengthwise under shoulder so that you will be able to easily pull end of strings together and secure
Take half of the fig and garlic mix and spread in opening on pork. Set aside remaining garlic and figs and their liquid.
Secure bakers twine so that roast is secure and your fig mix will not fall out during cooking.
Preheat oven to 325°. In a small bowl, combine 1 tablespoon thyme, 1 1/2 tsp. salt, 1/2 tsp. pepper. Rub roast with olive oil and then rub.
In dutch oven heat 2 tablespoons of butter and 2 tablespoons of olive oil over medium heat. Once butter is bubbling add shoulder and sear till brown on all sides. Should take 8-10 minutes total. (you are not cooking just browning the outside). Adjust heat as needed to keep meat from scorching.
Once browned transfer shoulder to a cutting board (I use lid of dutch oven turned upside to save on clean up)
Reduce heat to medium cook, add remaining butter and garlic stirring often, until garlic begins to turn light golden, about 1 minute.
Pour in remaining wine from bottle and bring to a boil, scraping up browned bits with a wooden spoon.
Add pork shoulder back into dutch oven, cover and place in oven.
Bake pork until almost tender when pierced, 2 1/2 hours.
Stir reserved fig mixture into Dutch oven with shoulder and bake, covered, until meat is tender, 30 minutes more. Pork should be falling apart.
Transfer to cutting board and let rest
Place pan with juices on stove top and boil over high heat until reduced to 2 cups, about 5 minutes. Stir in 1/2 teaspoon thyme.
Taste and season with lemon juice and more salt and pepper if needed.
You can serve sauce as is or if you want more of a gravy texture puree in a blender and serve in a gravy boat.
Remove twine from pork, then pull meat apart and serve on platter aside gravy