In a large pot of salted water, cook pasta until al dente. Drain, toss with a bit of olive oil and set aside.
Meanwhile, preheat large skillet on medium heat and swirl 1 tablespoon of olive oil to coat. Season chicken breasts with 1/2 teaspoon salt and 1/2 teaspoon pepper and pan fry until golden brown or for about 6 minutes per side. Transfer onto a plate and set aside.
Return skillet to medium heat and add remaining 1 tablespoon of olive oil. Add onion and cook until softened, stirring occasionally. Add garlic and saute for another 30 seconds, stirring a few more times.
Add chicken broth, milk, heavy cream, parmesan cheese, remaining 1/2 teaspoon salt, 1/4 teaspoon pepper and stir. Bring sauce to a boil, reduce heat to low and let it simmer for about 3-4 minutes, or until thickened a bit. Add lemon zest, lemon juice and give sauce another quick stir. Turn off heat.
Slice chicken into bite sized pieces and add to the skillet along with previously cooked pasta. Give it a gentle toss until lemon chicken pasta is well mixed. If your skillet isn't large enough, mix everything in a pot you cooked pasta in.
Let pasta stand for a few minutes for sauce to thicken and give another gentle toss. Garnish with parsley and parmesan cheese, and serve hot.