For the hazelnut pesto: Place the parsley, hazelnuts, Parmesan and garlic in a food processor and process until coarsely chopped. With the motor running, slowly add the olive oil and process until combined; transfer to a bowl and season with salt and pepper.
For the portobello salad: Heat a charcoal or gas grill to medium-high for direct and indirect grilling. Brush the mushrooms on both sides with 1/4 of the cup olive oil and season with salt and pepper. Place the mushrooms on a baking sheet and roast until cooked through, approximately 10 minutes on the grill. Remove to a baking sheet and loosely tent, reserving the liquid that the mushrooms have exuded.
Whisk together the reserved mushroom liquid, the sherry vinegar, mustard, remaining olive oil and season with salt and pepper.
Slice the mushrooms into 1/2-inch thick slices and place in a large bowl with the arugula, fennel, and some salt and pepper. Drizzle some of the vinaigrette down the sides of the bowl and toss the salad to combine. Spoon a pool of pesto on one side of a large platter. Mound the salad on the other side, and garnish everything with fennel fronds.
Place the mushrooms on a large platter, place the fennel over the mushrooms and drizzle with the pesto. Arrange the arugula around the platter.