Melt 2 tablespoons of French butter in a pan, then add the onion and sea salt and cook until golden brown.
Slide the onions to one edge of the pan and add the brown sugar and 1 tablespoon of soy sauce to the center, cooking until the mixture thickens into a glaze.
In another pan, melt 2 more tablespoons of French butter and add the shiitake mushrooms, cooking until they reduce by about half. Add the other tablespoon of soy sauce and cook just until thickened.
Toss the arugula with the dill, juice from the lemon, black pepper and the Parmigiano Reggiano cheese.
In another bowl, mix the mayonnaise with Dijon mustard.
Toast the sourdough bread with the remaining butter. Spread the mayonnaise-Dijon mixture to the interiors of the slices, then add the caramelized onions and soy-glazed buttered mushrooms. Top with the greens and the second slice of bread.
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