Season the boneless, skinless chicken breasts lightly with black pepper and set them aside.
In a medium bowl, whisk together the cream of chicken soup, low-sodium chicken broth, softened cream cheese, dry Italian dressing mix, and minced garlic until mostly smooth.
Place the chicken breasts in the bottom of the slow cooker in an even layer.
Pour the creamy sauce mixture evenly over the chicken and place the unsalted butter on top.
Cover and cook on low for 4 to 6 hours or on high for 2 1/2 to 3 1/2 hours, until the chicken reaches 165°F and is very tender.
About 15 minutes before serving, cook the angel hair pasta according to the package directions until al dente, then drain well.
Shred the cooked chicken gently with two forks or leave it in larger pieces, stirring it into the sauce until everything is well coated.
Serve the creamy angel chicken over the hot angel hair pasta and garnish with chopped fresh parsley or grated Parmesan if desired.