Caramelize the Onions: Meanwhile, melt the butter in a large skillet over medium heat. Add onions; saute until softened, then turn the heat down to caramelize the onions for 20 minutes or so, stirring occasionally until they are deep golden brown.
Make the Au Jus: Add the wine to the onions in the pan. Let it sizzle out. Add the Worcestershire and soy sauce. Add the broth. Simmer until reduced slightly. Pull out the onions with tongs and set aside. Season the au jus to taste.
Toast the French Dips: Butter the bread if you want. Pile mushrooms on the bread. Cover with cheese. Return to the oven on the same roasting pan for about 5 minutes (enough to toast the bread and melt the cheese).
Feast: Top the sandwiches with caramelized onions. Give em each a dollop of horseradish aioli. Serve with the au jus for dipping! HOLY MOLY that’s good.
