Cook pasta according to package directions. Reserve 1 cup pasta water. Drain and set aside.
Season chicken with salt and pepper..
Heat olive oil in a large skillet. Once hot, add one tablespoon of butter. Cook chicken is golden-brown and fully cooked; about 6-8 minutes per side (depending on the thickness of your chicken breast).*
Transfer cooked chicken onto a plate and cover with aluminum foil to keep warm.
Add the remaining butter to the skillet. Once melted add garlic and stir until fragrant. Add milk and cream and bring to a simmer.
Whisk in Parmesan and Mozzarella cheeses. Stir until the cheeses have melted and the sauce becomes thick and creamy. If your sauce looks a little too thick, add some of the reserved pasta water. Season to taste with salt and pepper.
Remove the sauce from heat and add in the cooked fettuccine. Use a set of tongues to combine the pasta with the sauce. Make sure the pasta is well coated.
Slice chicken breast and add to the pasta. Garnish with parsley, serve and enjoy.
The internal temperature for chicken is 165º. I like to check this with a meat thermometer.To grate the cheese using a simple grater like this one or if you want really finely grated cheese you can also use a Microplane.