Set a rack in the center of the oven and heat the oven to 450°F. Heat 2 Tbs. of the oil in a heavy, ovenproof 12-inch skillet over medium-high heat until shimmering hot. Add the mushrooms, sprinkle with 1/2 tsp. each of salt and pepper, and cook, stirring, until browned and tender, about 3 minutes. Stir in the spinach, sprinkle with salt, and cook, tossing well with tongs, until just wilted, about 2 minutes. Transfer the spinach mixture to a colander and set the skillet aside.
Trim the pork of any silverskin or excess fat. Butterfly the tenderloin by making a horizontal slice lengthwise through the tenderloin almost all the way to the other side. Open the meat flat, like a book. Cover with plastic wrap, and using a meat mallet, a small, heavy skillet, or the heel of your hand, lightly pound the pork so that it’s 1/4 inch thick. Rub the pork all over with 1 Tbs. of the oil, half of the thyme, and about 1/2 tsp. each of salt and pepper.
Squeeze any excess liquid from the spinach and mushrooms. Spread over the pork, leaving bare a 2-inch border along one long edge. Sprinkle on the Parmigiano. Starting with the long side that’s covered with filling, roll the stuffed tenderloin toward the bare-border side so that it forms a cylinder, and secure it with four or five toothpicks or kitchen twine.
Wipe the skillet clean if necessary. Heat the remaining 2 Tbs. oil in the skillet over medium-high heat until shimmering hot. Sear the pork on all three non-seam sides until well browned, about 6 minutes total. Flip onto the seam side, then transfer the skillet to the oven. Cook until an instant-read thermometer inserted into the thickest part of the roast registers 140ºF, 10 to 15 minutes.
Transfer the meat to a clean cutting board, tent with aluminum foil, and let rest for 10 minutes. Meanwhile, return the skillet to the stove over medium-high heat (be careful; the skillet’s handle will be hot). Add the shallots, season with 1/4 teaspoon salt, and cook, stirring, until the shallots soften and brown, about 2 minutes. Add the chicken broth, sherry vinegar, and the remaining 1-1/2 teaspoons thyme, and simmer briskly until the mixture reduces by a bit more than half, about 4 minutes. Stir in the cream and season with salt and pepper to taste.
Slice the pork into 1/2-inch-thick slices (a serrated blade works well) and serve with the sauce.
The robust flavors of a Grenache blend from southern France would be delicious with the creamy sauce and earthy filling.