Place dry mushrooms in a heatproof bowl and pour boiling broth over them. Let stand to soften, about 30 minutes. Line a small strainer with a coffee filter and strain mushroom liquid through filter and into a bowl or measuring cup. Squeeze mushrooms to extract any excess liquid, straining into a container to reserve liquid. Rinse reconstituted dried mushrooms to remove any grit, pat dry with paper towels and finely chop.
In a large skillet, heat a tablespoon olive oil and a tablespoon of butter over medium-high heat. Season pork on all sides with salt and pepper. Add it to pan and sear, turning it to brown on all sides, 3-4 minutes. Reduce heat to medium, cover and cook until medium-rare and internal temperature is 145-150 degrees F, 10-12 minutes. Transfer to a warmed platter and loosely cover with foil to keep warm.
Heat a half tablespoon olive oil and half tablespoon of butter in a medium skillet over medium-high heat. Add shiitakes and cook until they release their juices and are tender and lightly browned, about 5 minutes. Season with salt and pepper, remove from heat and set aside. Heat remaining half tablespoon olive oil and half tablespoon of butter in a small saucepan over medium heat. Add shallot and cook until softened, about 5 minutes.
Add wine, reserved mushroom liquid and chopped figs, bring to a boil and reduce by half. Add wine reduction to skillet used to cook pork. Bring to a boil, scraping the pan with a wooden spoon to remove any brown bits. Stir in thyme and demi-glace, if desired. Add chopped porcini mushrooms. Remove pan from heat and swirl in the remaining tablespoon butter. Salt and pepper to taste. Cut pork into 1-inch thick medallions and return them to pan to heat through. Briefly return pan with mushrooms to heat to warm. Divide the medallions among four dinner plates. Spoon fig and mushroom sauce on top of medallions and top each portion with a spoonful of shiitakes.
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