Step 1

Melt 1 tablespoon of the butter in a large frying pan over medium heat. Add the onion and sauté until very tender and slightly golden. Transfer to a medium bowl and set aside.

Step 2

Melt 2 tablespoons of the butter in the same pan over medium-high heat. Add a third of the mushrooms, stirring to coat in the butter. Cook, stirring occasionally, until the mushrooms are tender and golden, 5 to 7 minutes. (They may look dry at first—don’t worry!) Add 1/2 teaspoon each vinegar and smoked paprika, stirring well to combine, then transfer the mushrooms to the bowl with the onions. Repeat with the remaining butter, mushrooms, vinegar, and paprika in two more batches.

Step 3

Return the mushroom and onion mixture to the pan over medium-low heat. Add the barbecue sauce and water or broth, stir to combine, and cook until slightly thickened. Taste and season with salt and pepper as needed.

Step 4

To serve, toast or warm the brioche buns. Divide the filling between the buns and top with creamy slaw.