Put chicken leg quarters, onion, garlic, the 2 cups broth, salt and pepper in a 6-quart slow cooker. Cover and cook on low for 6 hours until the chicken reaches 165°F and the meat separates easily from the bones.
Lift chicken onto a tray and remove every bone, tendon and tough piece, keeping the meat in large bite-size pieces. Strain the cooking liquid, discard the spent onion and garlic and measure 3 cups, adding extra broth if needed.
Return measured liquid to the cooker, turn to high and stir in milk. Add dry fettuccine in small crossed handfuls, press into the liquid, cover and cook for 30 to 45 minutes, gently separating the strands every 10 to 15 minutes, until just tender.
Return boneless chicken, cover for 5 minutes and check that it reaches 165°F. Switch off and stir in cream and butter until the butter melts.
Add Parmesan gradually, folding gently until the sauce coats the noodles. Rest for 3 minutes and divide among six bowls, serving immediately rather than leaving the finished pasta on warm.