Step 1Combine the basil leaves, vinegar, garlic, mustard, salt and pepper in a mini-processor. Add the oil; mix well. (This can be made a few hours ahead and left at room temperature.) Set aside 3 tablespoons plus 2 teaspoons for the mushrooms and onions.
Step 1Heat the oven to 375 degrees.
Step 2Brush both sides of the mushrooms with 2 tablespoons plus 2 teaspoons of the reserved vinaigrette. Arrange the mushrooms in a single layer on a baking sheet. Place the onion slices in a bowl; toss with the remaining reserved tablespoon of vinaigrette. Mound them on the baking sheet with the mushrooms.
Step 3Bake until the mushrooms are tender and the onions are cooked down but still retain some texture, about 25 to 30 minutes. Set them aside until cool enough to handle, then cut the mushrooms on the bias into 4 to 5 slices. Coarsely chop the onions. (This can be done several hours ahead; keep at room temperature.)
Step 4To assemble, brush some vinaigrette on the bread. Divide the onions evenly and spread them on 4 slices. Arrange the arugula on each to cover, dividing evenly. Layer the cheese over each. Fan each portabello, overlapping, over the cheese. Season with salt, pepper and red pepper flakes. Close the panini. (This can be done several hours ahead.)
Step 5Spray oil on a large grill pan or frying pan. Heat until hot over medium heat; lightly spray oil on the outer surfaces of the panini. Cook, covered, until browned on one side, 3 minutes; turn, gently compress, then brown the other side, about 3 minutes more.
Step 6Let rest 5 minutes before cutting in half. Serve hot.