Preheat a large non-stick skillet over medium high heat with 2 Tablespoons of butter and caramelize the onions until deep golden brown, stirring occasionally for about 20 minutes. Remove the onions and set aside in a bowl. Add in the remaining butter and saute the mushrooms, being sure not to overcrowd them – work in batches if necessary – for about 5 minutes. Pour the onions back into the skillet with all the mushrooms, and add in the garlic, Worcestershire sauce (vegan if desired), soy sauce and toss them to combine. Pour the whole batch into a strainer set over a bowl.
In the same skillet (do not clean it), pour in the vegetable broth, and any juice that drips away from the mushrooms and onions. Bring the mixture to a simmer and reduce it by half. Serve it alongside the sandwiches.
To make the Sriracha Garlic Mayonnaise (aioli), in a small bowl, whisk together all the ingredients. Slater on toasted baguette, and top with the mushrooms and onions. Serve with a small dish of the “au jus” on the side.