How to Make It
Heat butter and 1 1/2 teaspoons of the oil in a large skillet over medium. Add onions; cook 10 minutes, stirring occasionally. Reduce heat to medium-low; cook until golden, about 30 minutes, stirring occasionally. Stir in salt; cook 5 minutes.
Heat 1 tablespoon of the oil in a separate large skillet over medium-high. Arrange half of mushrooms in skillet in a single layer and cook, undisturbed, 3 minutes, until golden-brown. Stir and cook 5 minutes, stirring often, until both sides are golden-brown. Place mushrooms in a bowl. Repeat process with remaining 1 tablespoon oil and half of mushrooms.
Combine all cooked mushrooms in pan. Add broth, sherry, Worcestershire sauce, thyme, soy sauce, and black pepper. Bring mixture to a boil; reduce to a simmer; cook 5 minutes. Use tongs to remove mushrooms from au jus; place in a bowl. Pour au jus evenly into 4 ramekins.
Combine mustard and horseradish in a bowl. Spread on top halves of rolls.
Preheat broiler to high. Arrange 3/4 cup mushrooms onto the bottom half of each roll. Add 1/4 cup onions. Tear each slice of cheese in half; place over onions. Place hoagies on a baking sheet. Broil 1 to 2 minutes, until cheese melts. Serve with au jus.