How to Make It
Heat butter and 1 1/2 teaspoons oil in a large skillet over medium. Add onions; cook, stirring occasionally, 10 minutes. Reduce heat to medium-low; cook, stirring occasionally, until golden, about 30 minutes. Stir in salt; cook 5 minutes.
Heat 1 tablespoon oil in a separate large skillet over medium-high. Arrange half of mushrooms in skillet in a single layer and cook, undisturbed, until golden brown, about 3 minutes. Stir and cook, stirring often, until golden brown all over, about 5 minutes. Transfer mushrooms to a bowl. Repeat process with remaining 1 tablespoon oil and remaining mushrooms.
Return cooked mushrooms to pan. Add broth, sherry, Worcestershire, thyme, soy sauce, and pepper. Bring mixture to a boil. Reduce to a simmer; cook 5 minutes. Using tongs, remove mushrooms to a bowl. Divide jus evenly among 4 ramekins.
Stir together mustard and horseradish in a bowl. Spread on top halves of rolls.
Preheat broiler to high. Arrange 3/4 cup mushrooms on bottom half of each roll. Top each with 1/4 cup onions. Tear each cheese slice in half; place two halves on each sandwich. Place sandwiches on a baking sheet; cover with top halves of rolls. Broil until cheese melts, 1 to 2 minutes. Serve with jus for dipping.