Pat chicken breasts dry and season both sides with salt and pepper.
Heat vegetable oil in a skillet over medium-high heat. Sear chicken for 2-3 minutes per side until golden brown.
In a medium bowl, whisk together cream cheese and butter until smooth.
Add cream of chicken soup, minced garlic, chicken broth, and Italian dressing mix. Stir until well combined.
Place seared chicken in the bottom of slow cooker. Pour sauce evenly over the top.
Cover and cook on LOW for 5-6 hours or HIGH for 3-4 hours until chicken reaches 165°F internally.
About 15 minutes before serving, cook angel hair pasta according to package directions.
Shred chicken directly in crockpot using two forks. Stir to coat in sauce.
Serve over pasta and garnish with fresh parsley.