Heat the olive oil in a medium skillet over medium high heat. Season the chicken with salt and pepper on both sides. Add the chicken to the skillet and cook 2-3 minutes per side, until browned but not cooked through.
Add the chicken to the crock pot. Mix together the whole milk, chicken broth, and garlic, and pour it over the chicken. Cook on high heat for 2-3 hours, or low heat for 4-5 hours.
Remove the chicken from the pot and cover with foil to keep warm. Add the fettuccine to the crock pot and stir to coat in the sauce. Cover and cook on high heat for 30 minutes, until tender. Stir in the heavy cream and parmesan cheese.
Slice the chicken and add it to the fettuccine in the crock pot. Serve sprinkled with parmesan and fresh parsley, if desired.