Step 1: In a skillet over medium heat the the butter and 1 1/2 teaspoons oil and then add the onions, cook, stirring for 10 minutes. Reduce heat to medium low and cook stirring until onions are golden (30 minutes) and then stir in the salt and cook another 5 minutes.
Step 2: In another skillet over medium high heat heat one tablespoon oil and then arrange half of the mushrooms in a single layer and cook until golden brown (do not stir or move the mushrooms) (3 -5 minutes). Stir and cook until golden brown all over for another 5 minutes. Transfer mushrooms to a bowl and repeat process with remaining oil (1 T) and mushrooms.
Step 3: Return cooked mushrooms to skillet and add next 6 ingredients (broth - pepper) and bring to a boil. Reduce to as simmer and cook for 5 minutes; using tongs remove mushrooms to a bowl and dived the jus among 4 ramekins.
Step 4: In a bowl combine the mustard and horseradish and spread over the top halves of the rolls.
Step 5: Preheat broiler to high. Arrange 3/4 cup mushrooms on bottom half of each roll, top with 1/4 cup onions. Tear each cheese slice in half and place two halves on each sandwich. Place sandwiches on cookie sheet; cover top halves of the rolls and broil until cheese melts (1-2 minutes). Serve with jus for dipping.
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