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1
Brush the underside of 4 of the Portobello mushrooms with olive oil.
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2
Cut the stems and the remaining 2 mushroom caps into 1/2-inch dice.
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3
Transfer the diced mushrooms to a food processor and pulse just until finely chopped.
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4
In a large skillet, heat the 2 tablespoons of olive oil until shimmering.
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5
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.
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6
Add the chopped mushrooms and cook over high heat, stirring, until softened and all of the liquid has evaporated, about 5 minutes longer.
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7
Add the cashews and half of the dill and transfer to a bowl to cool.
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8
Stir in the shredded cheese and season with salt and cayenne.
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9
In a small bowl, combine the mayonnaise with the remaining minced garlic and dill and season with black pepper.
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10
Light a grill.
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11
When the fire is medium hot, brush the grate with oil.
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12
Grill the mushrooms, stemmed side down, for 7 to 8 minutes, or until softened and browned.
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13
Brush the tops with olive oil and transfer them to a platter, stemmed side up.
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14
Pack the filling into the mushroom caps in a slightly flattened mound.
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15
Top with the cheese slices.
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16
Return the mushrooms to the grill, filling side up.
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17
Cover and grill over a medium-hot fire until the cheese is melted and the Portobello bottoms are browned and cooked through, 5 minutes.
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18
Grill the cut sides of the rolls until toasted.
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19
Spread 2 tablespoons of the mayonnaise on the rolls' cut sides.
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20
Set the mushrooms on the bottom halves, top with the arugula and tomatoes and serve, with the roll tops alongside.
