Ingredients

1. 6 large Portobello mushrooms (6 ounces each), stemmed and stems reserved
2. 2 tablespoons extra-virgin olive oil, plus more for brushing
3. 1 yellow bell pepper, finely chopped
4. 2 medium shallots, minced
5. 2 garlic cloves, minced
6. 1/3 cup coarsely chopped roasted salted cashews
7. 1/4 cup finely chopped dill or basil
8. 4 ounces aged Gouda or Cheddar cheese, shredded (1 cup), plus 4 thin slices aged Gouda or Cheddar
9. Kosher salt
10. Cayenne pepper
11. 1/2 cup mayonnaise
12. Freshly ground black pepper
13. 4 kaiser rolls, split
14. 1 small bunch of arugula
15. 4 thin tomato slices

Directions

1. Brush the underside of 4 of the Portobello mushrooms with olive oil. Cut the stems and the remaining 2 mushroom caps into 1/2-inch dice. Transfer the diced mushrooms to a food processor and pulse just until finely chopped. 
2. In a large skillet, heat the 2 tablespoons of olive oil until shimmering. Add the bell pepper, shallots and half of the garlic and cook over moderate heat, stirring, until softened and beginning to brown, about 5 minutes. Add the chopped mushrooms and cook over high heat, stirring, until softened and all of the liquid has evaporated, about 5 minutes longer. Add the cashews and half of the dill and transfer to a bowl to cool. Stir in the shredded cheese and season with salt and cayenne. 
3. In a small bowl, combine the mayonnaise with the remaining minced garlic and dill and season with black pepper. 
4. Light a grill. When the fire is medium hot, brush the grate with oil. Grill the mushrooms, stemmed side down, for 7 to 8 minutes, or until softened and browned. Brush the tops with olive oil and transfer them to a platter, stemmed side up. Pack the filling into the mushroom caps in a slightly flattened mound. Top with the cheese slices. 
5. Return the mushrooms to the grill, filling side up. Cover and grill over a medium-hot fire until the cheese is melted and the Portobello bottoms are browned and cooked through, 5 minutes. Grill the cut sides of the rolls until toasted. Spread 2 tablespoons of the mayonnaise on the rolls' cut sides. Set the mushrooms on the bottom halves, top with the arugula and tomatoes and serve, with the roll tops alongside.

Notes

Beer Recommendation: A complex, dry beer from Belgium that has a soft, malty fruitiness will blend with the earthy Portobello mushrooms, sweet peppers and fresh herbs here. Try the widely available Duvel, made by Moortgat.