Make the sauce reduction:
Melt 1 Tbs. of the butter in a 3- to 4-quart (preferably 8-inch-wide) saucepan over medium-low heat. Add the shallot and cook until softened, about 5 minutes. Add the port and the thyme sprigs and bring to a boil over high heat. Reduce the heat to maintain a brisk simmer and cook until the port has reduced to a syrupy texture, about 30 minutes—you should have about 1/2 cup including the shallot.
Meanwhile, bring the chicken broth to a simmer in a small saucepan. Off the heat, add the porcini and let them soak for 15 minutes. With a slotted spoon, transfer the porcini to a small bowl. Strain the soaking liquid through a fine sieve lined with a paper towel or coffee filter.
Add the soaking liquid and half of the porcini (about 1/3 cup) to the port reduction (save the remaining porcini for the stuffing). Bring the sauce to a boil over high heat. Reduce the heat to maintain a brisk simmer and cook until it has reduced to 1-1/3 cups, about 15 minutes. Strain the sauce through a fine strainer, pressing on the solids. You should have about 1 cup sauce. Set aside until the roast is done.
Make the stuffing:
Soak the currants in hot water for 10 minutes. Drain and put them in a food processor. Add the reserved soaked porcini. Heat 1 Tbs. of the oil in a 10-inch skillet over medium heat. Pat the chicken liver dry and cook on both sides until browned on the outside and just a little pink inside, 3 to 4 minutes total. Transfer to a plate, let cool slightly, and then add to the food processor. Process until finely chopped.
Heat the remaining 1 Tbs. oil in the skillet over medium-high heat. Add the mushrooms, salt, and pepper. Cook, stirring occasionally, until they look wilted and shrunken, 2 to 3 minutes. Add the shallot and garlic and continue to cook, stirring, until the mushrooms are tender and beginning to brown, 3 to 4 more minutes. Add the chicken broth and scrape the bottom of the pan with a wooden spoon to loosen any brown bits. If the liquid doesn’t evaporate right away, boil until it does. Remove the pan from the heat and let cool slightly.
Add the sautéed mushrooms, butter, parsley, Cognac or Armagnac, lemon juice, thyme, and lemon zest to the liver mixture in the food processor. Pulse to form a chunky paste and season to taste with additional salt and pepper.
Stuff & cook the roast:
Trim the silverskin and excess fat from the tenderloin. Butterfly the tenderloin by slicing it lengthwise almost but not completely in half, so that you can open it like a book.
Finish the sauce:
While the roast rests, bring the sauce to a simmer in a small saucepan over medium-low heat. Mix the softened butter with the flour to form a paste and whisk it into the sauce. Simmer the sauce to thicken slightly and cook off any raw flour taste, about 3 minutes. Whisk in the vinegar. Season to taste with salt, pepper, and additional vinegar.
Plate & serve:
Remove the strings from the roast and slice it into 8 to 10 medallions. Put a medallion on each plate and drizzle the sauce around the beef.