Preheat the oven to 300 degrees F. Butter a glass baking dish large enough to hold the burgers.
Sprinkle the meat with salt and pepper and form it into 4 patties. Place them into the prepared baking dish and bake until cooked through, 25 to 30 minutes. When they are done, set aside and keep warm. Add a little bit of wine to the pan and scrape up all the browned bits on the bottom. Set aside.
While the burgers are cooking, melt 2 tablespoons of the butter in a medium skillet over medium heat. Fry the mushrooms until they are just tender. Remove from the pan and keep warm.
Add the shallots to the pan with another tablespoon of butter and cook until they are softened. Pour in the wine and reduce by half. Pour in the juices from the baking dish and reduce again. Strain into a small pitcher and keep warm, reserving the shallots. Put the pan back on the heat and melt a tablespoon of butter. Add the pate slices and cook until just warmed through, 1 to 2 minutes.
Add the remaining 2 tablespoons butter to another skillet and toast the buns in the buttery juices.
To build the burger, place a mushroom cap on top of a burger bun. Set a burger on top, a spoonful of the reserved shallots and then a slice of pate. Pour some of the red wine reduction over the whole thing and top with the bun. Repeat with the remaining burgers. Serve with Truffled Oven Fries.
Truffled Oven Fries:
Preheat the oven to 425 degrees F.
Peel the potatoes and cut them into 1/2- by 1/2-inch strips. Toss them with the olive oil, season with salt and spread them in a single layer onto a baking sheet. Bake, tossing once or twice, until golden brown and crispy, 25 to 30 minutes.
Put the hot fries into a large bowl, drizzle with truffle oil and season with salt. Toss well to coat and serve immediately.