Heat olive oil in a large skillet on medium-hot heat. Generously season sliced chicken with salt and pepper, add to the hot skillet and cook for about 5 minutes until done. Remove sliced chicken from the skillet.
To the same skillet, add minced garlic, lime juice and chicken broth. Cook for 1-2 minutes on high heat, deglazing and scraping the skillet until the liquid is reduced and making sure garlic does not burn and brown too much. Mix in chopped cilantro.
Add heavy cream and milk. Bring to boil, the immediately add cheese and mix in until melted, while boiling. Then, remove from heat.
Cook pasta according to package instructions, drain, reserving some cooked pasta water.
Add pasta to the sauce, along with the half of the chicken. Mix everything in and season pasta with more salt, if needed.
Top with the remaining half of the chicken and more chopped cilantro.