Rinse chicken breasts in cold water dry off with paper towels. Cut chicken into bite size pieces. In a very large skillet heat two of the tablespoon. of olive oil on med high heat. Add the chicken to the pan. Season the chicken with salt and pepper to taste. Cook chicken until no longer pink inside and a little browned. Stir chicken occasionally (Drying the chicken before cooking will help with the browning). Should take about 5 minutes. Remove chicken from pan. Set aside.
Clean mushrooms with damp paper towel if needed (Do not rinse in water!). Most mushrooms these days are pretty clean so it should not take long to clean them. Chop mushrooms. When I chop mushrooms I make one small slice with the stem facing me. Then place flat side down so it will be easier to slice the mushroom without it sliding around on the cutting board. Slice lemons in half. When I juice lemons I put a small wire strainer over a bowl to catch the seeds. Take a fork, push it in the cut side of the lemon twist and squeeze. Set aside. Mince garlic cloves. Set aside.
In the same skillet that the chicken was cooked in, turn on medium to medium high heat and add the butter and remaining olive oil. Stir oil and butter until butter is melted. Add mushrooms, garlic, salt and pepper to taste. Reduce heat to simmer. Cook mushroom and garlic mixture for 10-12 minutes, stirring frequently. While mushrooms are cooking, add 4 quarts of cold tap water to a 6 quart pot. Cover pot and turn on high. When water starts to boil add 2 tablespoons of salt (This seems like a lot of salt but it is needed to flavor the pasta). Add the pasta and stir. Bring back to boil.
Cook for 8-10 minutes or until al dente (Pasta still has a little firmness to it, not completely soft). While pasta is cooking, add chicken to mushroom mixture. Stir. Add lemon juice stir and bring to a low boil. Turn down heat and simmer for 1-2 minutes. When pasta is done, drain (Do not rinse). Add pasta to skillet. Add 1 cup of the feta cheese. Toss until combined. Serve the extra ½ cup of feta on the side for sprinkling on top if desired.