Heat Butter and 1 1/2 tsp of the Oil in a large skillet over medium heat.
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 Tbsp of the Oil in a separate large skillet over medium-high heat.
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 Tbsp Oil and half of Mushrooms.
Combine all cooked mushrooms in pan. Add Broth, Sherry, Colonel Pabst 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 1-2 minutes, until cheese melts.
Serve with Au Jus.