Prepare a medium-hot charcoal fire or heat a gas grill for 20 minutes on high. Brush the mushrooms, chicken, scallions, and bread liberally with olive oil; season with salt and pepper. In a large bowl, stir the parsley, garlic, and 3 tablespoons of the olive oil together. Grill the mushrooms until flattened and golden brown (15 to 30 minutes), the chicken until streaked golden brown and springy to the touch but still moist inside (10 to 12 minutes), the scallions until slightly blackened (about 5 minutes), and the bread until golden brown (about 5 minutes), turning everything as you grill.

Meanwhile, bring a large pot of well-salted water to a vigorous boil and add the pasta. Cook until al dente; drain, reserving about 1 cup of the pasta liquid for tossing with the pasta. Slice the grilled mushrooms and chicken thinly to about the same size as the penne; chop the scallions into 1/2-inch lengths. Crumble the toasted bread. Add the pasta, chicken, and scallions to the dressing in the bowl and toss. Add 1/4 cup olive oil and about 1/2 cup of the reserved pasta water to moisten the pasta, using more or less water as needed. Finally, add the portabellas and the crumbled toasted bread. Season with salt and pepper. Serve with freshly grated Parmigiano, if you like.