Prep the slow cooker base: Add the onion, garlic, chicken broth, Italian seasoning, salt, pepper, and red pepper flakes to the crock. Stir to combine.
Add the chicken: Nestle the chicken breasts into the mixture.
Scatter the butter and cream cheese cubes over the top.
Cook low and slow: Cover and cook on Low for 3.5–4.5 hours, or on High for 2–3 hours, until the chicken is cooked through and easily shreds.
Shred the chicken: Remove the chicken to a plate and shred with two forks. Return it to the crock and stir to melt the cream cheese fully into the cooking liquid.
Add the cream and Parmesan: Pour in the heavy cream and stir in the Parmesan until smooth. Cover and cook on Low for another 15–20 minutes to thicken slightly.
Cook the broccoli: While the sauce finishes, steam the broccoli until crisp-tender (about 3–4 minutes for fresh; 2–3 minutes for frozen) or toss it directly into the crock for the last 20–30 minutes on Low if you prefer hands-off.
Cook the pasta: Boil the fettuccine in salted water according to package directions.
Reserve 1/2 cup pasta water before draining.
Combine: Stir the broccoli into the Alfredo chicken. Add the drained pasta and toss to coat. If the sauce is too thick, loosen with a splash of pasta water until silky.
Taste and finish: Adjust salt and pepper.
Squeeze a little lemon over the top for brightness if you like. Garnish with parsley and extra Parmesan.
Serve: Spoon into warm bowls. The sauce will continue to thicken as it sits, so serve promptly for the creamiest texture.