Preheat the oven to 425 degrees F (220 degrees C).

Place beef tenderloin in a baking dish. Spread 2 tablespoons softened butter over beef.

Bake in the preheated oven until browned, 10 to 15 minutes. Remove beef from the pan and reserve pan juices; allow beef to cool completely.

Increase oven temperature to 450 degrees F (230 degrees C).

Melt 2 tablespoons butter in a skillet over medium heat. Sauté onion and mushrooms in butter for 5 minutes. Remove from heat and let cool.

Mix paté and remaining 2 tablespoons softened butter together in a bowl; season with salt and pepper. Spread paté mixture over beef. Top with onion and mushroom mixture.

Roll out puff pastry dough to about 1/4-inch thickness and place beef in the center.

Fold up and seal the edges, ensuring seams are not too thick. Place beef in a baking dish, cut slits in the top of dough, and brush with egg yolk.

Bake in the oven for 10 minutes. Reduce heat to 425 degrees F (220 degrees C) and continue baking until pastry is golden brown, 10 to 15 minutes. Check for medium rare (122 to 130 degrees F). Rest for a few minutes.