In a pan, heat olive oil and sauté the shallots until translucent.
Add the mulled wine and balsamic vinegar. Cook until the liquid has almost completely evaporated.
Add the chopped mushrooms and garlic. Sauté until the mushrooms are cooked through.
Stir in the lemon zest and chopped thyme. Season with salt and freshly ground white pepper. Let the mixture cool completely.
Using a small filleting knife, cut a pocket into each steak. Keep the opening as small as possible while creating a large cavity inside by moving the knife tip in a half-circle motion.
Transfer the cooled filling into a piping bag and fill, alternatively use a small spoon to push the stuffing into the steaks. Seal the openings with toothpicks.
Sear the stuffed steaks in a medium hot pan with oil.
Just before they are fully cooked, add the vegan butter, rosemary, and garlic to the pan. Finish with salt and pepper.
Remove the toothpicks before plating, serve with your favorite side dish and sauce.