How to Make It
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.
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.
In a small bowl, combine the mayonnaise with the remaining minced garlic and dill and season with black pepper.
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.
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.