In a large pot, bring 6 quarts of salted water to a boil.
In a large saute pan over high heat, add 2 tablespoons olive oil. Season turkey with salt and pepper. Add turkey and saute breaking any large lumps until golden brown. Remove meat from pan and set aside. Add 2 tablespoons olive oil to pan and heat. Add chopped mushrooms and season with salt and pepper. Saute until all the liquid from the mushrooms has evaporated, about 5 minutes. Add the peas and saute for 4 minutes. Return the meat to the pan and cook for another 3 minutes. Remove from heat and set aside.
When the pot of water has come to a boil, add pasta and cook until al dente, about 8 minutes. Drain in a colander. Return pasta to the pot and add the meat mixture. Combine thoroughly and gently cook over medium heat for 5 minutes. Drizzle in remaining olive oil. If the mixture is dry, do not hesitate to use more olive oil if necessary. Check the seasoning. Turn off the heat and add the grated cheese. Stir well and serve in large bowl.
Grana Padano is a hard, grainy cow's milk cheese similar in flavor to Parmigiano-Reggiano.