For the walnut pesto: In a food processor, combine the arugula, basil, Parmesan, walnuts, garlic, lemon zest and juice, 1 teaspoon salt and 1/2 teaspoon pepper and pulse to blend. With the machine running, pour in the olive oil in a steady stream and process until smooth, stopping to scrape down the sides of the bowl as needed. Taste and adjust the salt and pepper if needed. Set aside. For the pasta: Bring a large pot of water to a boil and add salt. Add the pasta and cook until tender, stirring occasionally, about 11 minutes or according to the package directions. Drain the pasta, reserving 1 cup of the cooking liquid.
In the meantime, heat a large saute pan over medium heat. Melt 1 tablespoon butter, add the olive oil and then stir in the sliced onions. Season with salt and pepper and cook, stirring frequently, until the onions begin to brown or caramelize, about 10 minutes.
In the same pan, melt the remaining tablespoon of butter and add the mushrooms. Cook, stirring occasionally, until tender and brown, about 5 minutes.
Add the drained pasta to the pan, along with the walnut pesto, Parmesan, lemon juice and reserved cooking liquid. Season with salt and pepper. Garnish with the roasted red peppers, lemon zest and Parmesan.